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CPA

CERTIFIED PUBLIC ACCOUNTANTS

PART III

SECTION 5

ADVANCED FINANCIAL MANAGEMENT

STUDY TEXT

Revised on: July 2019

KASNEB JULY 2018 SYLLABUS

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SYLLABUS PAPER NO.15 ADVANCED FINANCIAL MANAGEMENT

GENERAL OBJECTIVE This paper is intended to equip the candidate with knowledge, skills and attitudes that will enable

him/her to apply advanced financial management techniques in an organisation.

15.0LEARNING OUTCOMES A candidate who passes this paper should be able to:

- Evaluate advanced capital budgeting decisions

- Design an optimal capital structure for an organisation

- Predict corporate failure

- Apply derivatives in financial risk management

- Apply financial management skills in the public sector

- Understand concepts of corporate restructuring and re-organisation

- Apply valuation techniques in real estate finance

CONTENT

15.1Advanced capital budgeting decision

- Incorporating risk/uncertainty in capital investment decisions

- Nature and measurement of risk and uncertainty

- Techniques of handling risk: sensitivity analysis, scenario analysis, decision trees, simulation

analysis, utility analysis, risk adjusted discounting rate(radr) and certainty equivalent method

- Incorporating capital rationing in capital investment appraisal

- Incorporating inflation in capital investment appraisal

- Evaluation of projects of unequal lives

- The real options-strategic investment option, timing option, abandonment option and the

replacement option

- Common capital budgeting pitfalls

15.2Portfolio theory and analysis:

- The modern portfolio theory: background of the theory; portfolio expected return; the actual

and weighted portfolio risk; derivation of efficient sets; the capital market line (CML) model

and its applications, the mean variance dominance rule; short comings of portfolio theory

- Capital Asset Pricing Model-CAPM : background of the theory; assumptions; beta estimation

- beta coefficient of an individual asset and that of a portfolio and the interpretation of the

result; security market line(SML) model and its applications; conceptual differences between

portfolio theory and capital asset pricing model

- Shortcomings of the capital asset pricing model

- The Arbitrage pricing model (APM) and other multifactor models: background of the theory;

conceptual differences between the Capital asset pricing model and the Arbitrage pricing

model; application of the Arbitrage pricing model, shortcomings of Arbitrage pricing model;

Pastor Stambaugh model

- Evaluation of portfolio performance: Treynor’s measure, Sharpe’s measure, Jensen’s

measure, appraisal ratio measure, information ratio, Modigliani and Modigliani (M2)

15.3Advanced financing decision

- The nature of financing decision, principle objectives of making financing decision

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- Overview of cost of capital: meaning and relevance of cost of capital: the firm’s overall cost

of capital; weighted average cost of capital (WACC) and weighted marginal cost of capital

(WMCC) ; analysis of breakpoints in weighted marginal cost of capital schedule

- Capital structure theories: nature of capital structure and factors influencing the firm’s capital

structure; traditional theories of capital structure - assumptions of the theories, Net income

theory and Net operating income theory; Franco Modigliani and Merton Miller’s propositions

- MM without taxes, MM with corporation taxes, MM with corporation and personal tax rates

and MM with taxes and financial distress costs; other theories of capital structure; the pecking

order theory and Trade-off theory determination of the firm’s optimal capital structure using

the Hamada model, CAPM and WACC

- Special topics in financing decision: analysis of operating profit (EBIT)/EPS at point of

indifference in firm’s earnings; establishing the range of operating profit within which each

financing option; leverage and risk; operating leverage and operating risk, financial leverage

and financial risk, combined leverage and total risk; quantifying leverage using the degree of

operating leverage, degree of financial leverage and degree of combined leverage

- Long term financing decisions; bond refinancing decision, lease-buy evaluation and the rights

issues

- Impact of financing on investment decisions - the concept of adjusted present value (APV)

15.4Mergers and acquisitions

- Nature of mergers and acquisitions

- Reasons of mergers and acquisitions

- Acquisition and Mergers verses organic growth

- Valuation of acquisitions and mergers

- Prediction of a takeover target

- Defence tactics against hostile takeovers

- Financing of mergers and acquisitions

- Analysis of combined operating profit (EBIT) and post-acquisition earning per share at the

point of indifference in firms earnings under various financing options.

- Determination of range of combined operating profit.

- Regulatory frame work for mergers and acquisitions

- Reasons why there are failed mergers and acquisitions

- Mergers and acquisitions in a global context

15.5Corporate restructuring and re-organisation

- Background on restructuring and re organisation

- Indicators/symptoms of restructuring

- Considerations in designing an appropriate restructuring programme

- Financial reconstruction: forms of financial reconstruction; impact of financial reconstruction

on share price; impact of financial reconstruction on the weighted Average cost of capital

(WACC)

- Portfolio reconstruction: various ways of unbundling a firm: divestment, de-merger, spin-off,

liquidation, sell-offs, equity curve outs, strategic alliances, management buyout, leveraged

buyouts and the management buy-ins.

- The relevance of the various forms of portfolio reconstruction

- Organisational reconstruction: The nature and benefits of this form of restructuring; models of

predicting corporate failure; Multiple discriminant analysis (Z-Score model), Beaver failure

ratio, Argenti model, Taffler’s model

- Causes of financial distress

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- Forms of financial distress and solutions to financial distress

15.6Derivatives in financial risk management

- The meaning, nature and importance of derivative instruments: futures, forwards, options and

swaps

- Pricing and valuations of derivatives: futures, forwards, options and swaps

- Types of risks: operational risks, political risks, economic risks, fiscal risks, regulatory risks,

currency risks and interest rate risks

- Foreign currency risk management: Types of forex risks, hedging currency risks, forward

contracts, money market hedge, currency options, currency futures and currency swaps

- Interest rate risks: Term structure of interest rates, forward rate agreement, interest rate

futures, interest rate swaps, interest rate options

15.7International financial management

- International investments

- International financial markets

- International financial institutions

- Methods of financing international trade

- International parity conditions: Interest rate parity, purchasing power parity and International

fisher effect

- International arbitrage: locational arbitrage, triangular arbitrage and covered interest arbitrage

- Divided policy for multinationals

- International debt instruments: International bonds (euro bond), certificate of deposits,

securitisation of loans, commercial paper

- Availability and timing of remittances

- Transfer pricing: impact on taxes and dividends

15.8Real estate finance

- Overview of real estate business - nature of real estate business, legal and economic

framework and participants in real estate business in Kenya

- Valuation approaches (income, cost and sales comparison approaches)

- REITS: types; advantages and disadvantages; valuation: net asset value per share (NAVPS);

use of funds from operations (FFO), adjusted funds from operations (AFFO) in REIT

valuation

- Instruments of real estate financing - mortgages, lien, title, mortgage requirements and

mortgage clauses

- Rights in case of debt - default and its consequence, equity of redemption, foreclosure,

statutory redemptions

- Mortgage and financial markets: demand for funds in mortgage market, disintermediation

effects, primary and secondary mortgage market, mortgage market and cost of money, role of

central bank and the role of government in mortgage markets

- Savings and loan association - classification, state accounts, insurers. Mortgage backed bonds

and services

15.9 Emerging issues and trends

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CONTENT PAGE

Topic 1: Advanced capital budgeting decision…………………………….…….…6

Topic 2: Portfolio theory and analysis…………………………………………..…69

Topic 4: Advanced financing decision……………………………………………112

Topic 4: Mergers and acquisitions………………………………………..…….…186

Topic 5: Corporate restructuring and re-organisation…………….………….……217

Topic 6: Derivatives in financial risk management……………………….………228

Topic 7: International financial management………………………………..……283

Topic 8: Real estate finance……………………………………………….…....…306

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CHAPTER ONE

ADVANCED CAPITAL BUDGETING DECISION

CHAPTER KEY OBJECTIVES

To be able to understand the following with regard to capital budgeting decisions;-

1. Incorporating risk/uncertainty in capital investment decisions

2. Nature and measurement of risk and uncertainty

3. Techniques of handling risk: sensitivity analysis, scenario analysis, decision trees, simulation

analysis, utility analysis, risk adjusted discounting rate(RADR) and certainty equivalent method

4. Incorporating capital rationing in capital investment appraisal

5. Incorporating inflation in capital investment appraisal

6. Evaluation of projects of unequal lives

7. The real options-strategic investment option, timing option, abandonment option and the

replacement option

8. Common capital budgeting pitfalls

1.1 INTRODUCTION

These decisions involve investing of a company’s funds in long term projects that are more beneficial

to the firm i.e. projects that aim at maximisation of shareholders’ wealth or value of the firm. It is the

process of determining viability of projects.

Capital investment refers to the real act of expenditure that involves allocation of capital/resources

the available company projects.

Capital Budgeting refers to the process of planning and evaluating the profitability/viability of the

available projects to be undertaken with an aim of implementing the most profitable i.e. the project

that would maximise the value of the firm/shareholders wealth.

Importance of capital budgeting.

Most projects involve a heavy investment of funds, which may lead to significant losses if not

properly planned.

Such decisions would affect the long-term growth of the firm. Profitable projects would increase

the value of the firm and the shareholders’ wealth, which is the main objective of any company.

Such decisions are largely irreversible and if reversed it will be at a substantial loss.

1.2 INCORPORATING RISK/UNCERTAINTY IN CAPITAL INVESTMENT

DECISIONS

Investment Decisions under Uncertainty/Risk

Investment appraisal faces the following problems;

The decisions made are based on forecasted cash flows

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The forecasted cash flows are subject to uncertainty

This uncertainty has to be reflected in financial evaluations

Risk

This refers to a quantifiable possible outcome that has some associated probabilities because of the

past data that is available about such circumstances. Its the possibility of a firm’s earnings fluctuating

through time

Uncertainty

This refers to unquantifiable possible outcome that cannot be measured using mathematical models

techniques since it does not have any associated probability i.e. the investor has no past data of such

assurances e.g. A project being implemented for the first time.

Uncertainty is more difficult to plan, for obvious reasons. Uncertainty can be dealt with in project

appraisal in several ways.

In investment appraisal three areas of concern includes –

- The projects economic life

- Forecasted cash flows and their associated probabilities

- The discount factor/firms cost of capital

Three methods are available for use when incorporating risk in capital budgeting i.e.

1. Expected monetary value

2. Standard Deviation

3. Coefficient of variation

Expected Monetary Value

The expected value is a weighted average of the expected cash flows of a project.

It is determined by the value of cash flow and the associated probabilities.

Expected Value = ∑Return × Probability

The higher the Expected value, the lower the risk a project has.

Standard Deviation

This measures the spread of data around the expected value.

The higher the standard deviation, the hire the risk a project has since cash flows can deviate more

from the actual return.

Three methods are available in calculating the standard deviation (SD) depending on variables

provided.

1. When probabilities are provided;

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SD (𝛿) = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

𝛿=√∑(𝑟𝑒𝑡𝑢𝑟𝑛 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑𝑟𝑒𝑡𝑢𝑟𝑛)2 × 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦

2. In absence of probabilities and given a sample size of more than 30 (large sample)

𝛿 = √∑(𝑟𝑒𝑡𝑢𝑟𝑛−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑𝑟𝑒𝑡𝑢𝑟𝑛)

2

𝑁

N = Sample size

3. In absence of probabilities and given a sample size of 30 or less (small sample)

𝛿 = √∑(𝑟𝑒𝑡𝑢𝑟𝑛−𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑𝑟𝑒𝑡𝑢𝑟𝑛)

2

𝑁−1

Coefficient of Variation

It is a measure of dispersion of data that is calculated statistically

It is used in comparing the degree of variation from one data series to another.

It is a relative measure that indicates the amount of risk taken to generate a standard mean return.

The lower the coefficient of variation the lower the risk a project has and vice versa.

CV = Standard deviation

expected value x 100

CV = 𝜎

𝐸𝑅×100

Illustration 1

A project has the following possible outcomes, each of which is assigned a probability of occurrence.

Probability Present value

Sh.

Low demand 0.3 20,000

Medium demand 0.6 30,000

High demand 0.1 50,000

What is the expected value of the project?

Solution

The expected value is the sum of each present value multiplied by its probability.

Expected value = (20,000 × 0.3) + (30,000 × 0.6) + (50,000 × 0.1) = Sh.29, 000

Illustration 2

CV = Coefficient of variation

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What would happen to the expected value of the project above if the probability of medium demand

fell to 0.4 and the probability of low demand increased to 0.5?

Solution

Expected value = (20,000 x 0.5) + (30,000 x 0.4) + (50,000 x 0.1) = sh.27, 000

The project is riskier than before, as there is a greater probability of demand being low, which results

in a lower expected value.

Illustration 3

Tajiri Ltd is considering investment of sh.50, 000,000. The estimated annual net cash inflows over the

next five years under the three states of nature are as follows:

Project A

State of nature Probability Amount

Sh. “000”

Most pessimistic

Most likely

Most optimistic

0.25

0.50

0.25

13,500

18,000

20,000

Concerns have been raised about the possibility that this project will infringe on a competitor’s patent.

If this was the case and the competitor successfully pursued a claim for damages, the competitor may

have to be paid as much as sh.100, 000,000 in the third year. Lawyers estimate that there is only a 0.1

probability that this will happen.

Project B

This project will require an initial outlay of sh.50, 000,000 spread in equal instalments over the next

three years to finance a research project. If this project is successful and there is a probability of 0.5 of

this happening, it will lead to issuance of a patent right with an estimated value at the end of the end

of the three years of sh.200, 000,000. If not successful, the whole of the expenditure would have to be

written off.

Project C

This project will have an initial cost of sh.20, 000,000 and is expected to yield annual cash flows of

sh.8,000,000 in each of its first two years. Thereafter, the outcome is so uncertain that no estimate can

be given.

The company’s cost of capital is 14% per annum.

Required;-

Advise Tajiri Ltd on whether they should undertake the projects above.

Solution

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Project A

Initial outlay = Sh. 50,000,000

Expected annual cash flows = ∑ 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 × 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦

= 13.5m × 0.25 + 18m × 0.50 + 20m×0.25= Sh. 17,375,000

The cash flow above is an annuity and therefore use PVIFA for discounting.

Net Present Value= PV of Cash flows – Initial Outlay

17,375,000 × PVIFA14%5yrs – 50,000,000

17,375,000×3.4331 – 50,000,000 = Sh. 9,650,112.5

In case the competitor succeeds in the suit

NPV = 9,650,112.5 – (100,000,000 × PVIF3yrs14% × 0.1

9,650,112.5 - 100,000,000 × 0.6750 × 0.1

9,650,112.5 –6,750, 000

= Sh. 2,900,112.5

The project is viable since it has a positive NPV even after factoring the cost paid for the suit.

Project B

PV of initial outlay 50𝑚

3× PVIFA3yrs14% (1 + r)

50𝑚

3×2.3216 × 1.14 × 0.5

= Sh. (22.0552)

PV of Patents received: 200,000,000 × PVIF3yrs14%

200,000,000 × 0.6750 = Sh. 135m

NPV = 135m – 22.0552 = Sh. 112,944,800

Project B is viable if successful

Project C

End of year Cashflows

Sh.

Discount factor

14%

Present Value

Sh.

0

1

2

(20,000,000)

8,000,000

8,000,000

1.0000

0.8772

0.7695

(20,000,000)

7,017,600

6,156,000

= Sh.(6,826,400)

Project C has a negative NPV and therefore it is not viable.

N/B Discounting factors (PVIF) are read from the lump sum/irregular cash flows table.

The (1 + r) is included because the initial

payments are being done upfront

R = 0.14

1 + r = 1.14

PVIFA14%,5 = 3.4331 as read from annuity tables.

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1.3 ADVANCED TECHNIQUES OF HANDLING RISK

1. Sensitivity Analysis

The focus in this case is determining the effect on NPV due to a a change in a given decision variable

holding other factors constant. Variables are changed one at a time.

The concept of sensitivity analysis involves posing a question “what if” example

What if sales volume falls by 10%, will the NPV remain positive.

For investment decisions to change from Accept to Reject NPV should be less than 0 and the

variables must change by the sensitivity margin.

Sensitivity analysis is divided into two:

1) Breakeven point sensitivity analysis

2) Impact analysis

Weaknesses of sensitivity analysis

These are as follows;-

1. The method requires that changes in each key variable are isolated. However,

management is more interested in the combination of the effects of changes in two or

more key variables.

2. Looking at factors in isolation is unrealistic since they are often interdependent.

3. Sensitivity analysis does not examine the probability that any particular variation in costs

or revenues might occur.

4. Critical factors may be those over which managers have no control.

Illustration

R Ltd is considering a project with the following cash flows:

Year Cost of plant

Sh. “000”

Running costs

Sh. “000”

Savings

Sh. “000”

0

1

2

10,000

4,000

5,000

12,000

14,000

Cost of Capital = 9%

Required:

(i)Determine the sensitivity of the project to changes in the levels of cost of plant, running costs and

savings (considering each factor at a time) and assuming each factor is varied adversely by 10%

(ii)Comment on the factor which is most sensitive to adverse variations.

Solution

(i)

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Year Details Cashflows

Sh. 000

Discount factor

9%

Present value

Sh. 000

0

1

1

2

2

Cost of plant

Running costs

Savings

Running costs

Savings

(10,000)

(4,000)

12,000

(5,000)

14,000

1.0000

0.9174

0.9174

0.8417

0.8417

NPV

(10,000)

(3,669.6)

11,008.8

(4,208.5)

11,783.8

4,914.5

NPV if cost of plant is increased by 10%

NPV = 4914.5 – 10% ×10,000 ×PVIF0yrs9%

4914.5 – 1000 = 3,914.5

= Sh. 3,914.5

% change in NPV

4914.5−3914.5

4914.5 ×100%=20.35%

NPV if running costs increase by 10%

NPV = 4914.5 – (400 × PVIF1yr9% + 500 × PVIF2yrs9%

4,914.5 – (400 × 0.9174 + 500 × 0.8417)

4,914.5 – 787.81

= Sh. 4,126.69

% change in NPV

4914.5−4126.69

4914.5× 100% = 16.03%

NPV if savings reduce by 10%

NPV = 4,914.5 – (1,200×PVIF1yr9% + 1400 × PVIF2yrs9%

4914.5 – (1,200×0.9174 + 1,400 × 0.8417)

4,914.5 – 2,279.26 = Sh. 2635.24

% change in NPV

4914.5−2635.24

4914.5× 100% = 46.38%

Method 2

(ii) Savings are more sensitive to adverse variations than the other factors.

Scenario Analysis

A simple sensitivity analysis assumes that the variables are independent of each other.

Practically, this is quite impossible since most variables are interrelated such that a change in one

variable may lead to a change in another variable e.g. when sales increase by 10% the variable costs

are also expected to increase to sustain the increase in sales.

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Scenario analysis therefore considers all these variables changing simultaneously to give a particular

scenario to the managers.

This behavioural approach is used instead of sensitivity analysis to evaluate the impact of various

circumstances in decision-making.

It normally provides three different types of scenarios, that is

Worst Case Scenario

This is a pessimistic view of likelihood of future variables to change to the worst.

Base Case/Average Scenario

This represents the average and most likely of each variable. It represents what the analyst

believes that is most likely to occur.

Best Case Scenario

This is an optimistic view of the likely future events and it represents the best reasonable

estimates of each occurrence.

Illustration

Omena Ltd is a firm in the manufacturing industry. The management of this company is considering

purchasing a new machine at a cost of sh.125 million. This investment is expected to reduce

manufacturing costs by sh.45 million annually. The firm will need to increase its net operating

working capital by sh.12.5 million when the machine is installed, but the required operating working

capital will return to the original level when the machine is sold after 5 years.

Omena Ltd will use the straight line method to depreciate the machines and it expects to sell the

machine at the end of 5 years operating life for sh.11.50 million. The company pays corporation taxes

at the rate of 30% and uses 10% cost of capital to evaluate projects of this nature.

Required:

(a) The project’s net present value.

(b) The firm’s management are unsure about the annual savings in operating costs that will occur

with the new machines acquisition. Management believes that these savings may deviate from

their base case value (sh.45 million) by as much as a plus or minus 10%.

Determine the net present value of the project under both situations and comment on the

sensitivity of this variable.

Solution

Initial investment cost

Sh. M

Cost of machine

Investment in working capital

125

12.5

137.5

Annual after Tax Cashflows

Manufacturing costs are

tax allowable

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Sh. M

Decrease in manufacturing costs 45 (1 – 0.3)

Add depreciation tax shield [125−11.5

5] 30%

Annual net after tax cashflows

31.5

6.81

38.31

Terminal cashflows

Scrap value

Add release working capital

Sh. M

11.5

12.5

24.0

NPV = 38.31 x PVIFA5yrs10% + 24 x PVIF5yrs10% - 137.5

38.31 x 3.7908 + 24 x 0.6209 – 137.5

= Sh. 22.627148

NPV →

Best Case

Sh. “m”

Worst case

Sh. “m”

Annual cashflows

Dep. tax shield

Annual after tax

110% × 45 (1-0.3) = 34.65

6.81 6.81

41.46

90% × 45 (1-0.3) = 28.35

6.81

35.16

41.46 ×PVIFA5yrs10% +

24 × PVIF5yrs10% - 137.5

= Sh. 34.568168

35.16 ×PVIFA5yrs10% +

24 × PVIF5yrs10% - 137.5

= Sh. 10.686128

Best Case = 34.568168−22.627148

22.627148× 100 = 52.77%

Worst Case = 22.627148−10.868128

22.627148× 100 = 52.77%

The operating costs are sensitive to the NPV by 52.77% in both the Best Case and Worst Case

Scenarios.

Illustration 2

Suppose the firm’s chief finance officer suggest that the firm does a scenario analysis for a that costs

Ksh.125,000 and which will be scrapped after 5 years. The net operating working capital (NOWC)

requirement which will be released at the end of the project is shown below. After an extensive

analysis, she arrives with the following probabilities and values for the scenario analysis:

Scenario

Probability

Annual operating cost saving

Sh. ‘000’

Salvage

value

Sh. ‘000’

NOWC

Sh. ‘000’

Depreciation tax shield =

Depreciation ×Tax

Page 14

Worst case

Base case

Best case

0.4

0.4

0.2

36,000

45,000

54,000

9,000

11,500

14,000

15,000

12,500

10,000

Determine the projects expected net present value (ENPV), standard deviation and its coefficient of

variation.

Solution

NPV in Worst Case

Sh. “m”

Initial outlay

Cost of new machine

Add working capital investment

125

15

140

Annual cash flows (Savings in operating costs)

Operating cost saving 36 (1 – 0.3)

Add depreciation tax shield

[125−9

5] 30%

25.2

6.96

32.16

Terminal Cash flows

Scrap value

Release in working capital

9

15

24

NPV = 32.16×PVIFA5yrs10% + 24 x PVIF5yrs10% - 140

32.16 × 3.7908 + 24 × 0.6209 – 140

121.912128 + 14.9016 – 140 = -3.186272(Sh. millions)

NPV in Best Case

Sh. “m”

Initial outlay

Cost of new machine

Add working capital investment

125

12.5

137.5

Annual Cash flows

Operating cost saving 45 (1 – 0.3)

Add depreciation tax shield

31.5

Page 15

[125−11.5

5] 30% 6.81

38.31

Terminal Cash flows

Scrap value

Release of working capital

11.5

12.5

24

NPV = 38.31 x 3.7908 + 24 x 0.6209 – 137.5 = 22.627148

Expected Net Present Value = NPV x probabilities

-3.186272×0.4 + 22.627148 ×0.4 + 48.440568 × 0.2

Sh. “m” 17.464464

𝜎 = √∑(𝑁𝑃𝑉 − 𝐸𝑁𝑃𝑉)2

(-3.186272 – 17.464464)2× 0.4 =

(22.627148 – 17.464464)2×0.4 =

(48.440568 – 17.464464)2× 0.2 =

170.5811589

10.66132248

191.903038

373.1462849

𝜎 = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒

𝛿 = √373.1462849 = 19.32

Coefficient of variation = 𝜎

𝐸𝑁𝑃𝑉

Evaluated value = 𝜎

𝐸𝑁𝑃𝑉 =

19.32

17.46× 100 = 110.7%

Illustration 1

A company is considering undertaking a project that would cost Sh. 4m. It has an economic life of 5

years and a nil salvage value. Depreciation is to be charged on straight line basis. Tax is 30% and the

cost of capital is 12%. The following additional information relates to the project.

Variables Worst Possible outcome Best

Units produced and sold annually

Units selling price (Sh.)

Annual fixed costs (Sh.)

11,000

750

100,000

12,000

780

120,000

13,500

800

150,000

The company’s contribution margin is 85%

Page 16

Required;-

Calculate the base case NPV, worst case NPV and best case NPV of the project and comment on the

risk of the project.

Solution

Worst case

scenario

Sh.

Base case

scenario

Sh.

Best case

scenario

Sh.

Contribution margin (85% of selling price)

Units

Total annual contribution

Less fixed cost

Annual before tax cash flows

Less tax @ 30%

After tax cash flows

Add: Depreciation tax shield

30% x (Depreciation)

Net after tax cash flows (annuities)

Discount factor

Present value of cash inflows

Less initial outlay

637.5

11000

7,012,500

100,000)

6,912,500

(2,073,750)

4,838,750

240,000

_____

5,078,750

3.6048

18,307,878

(4,000,000)

14,307,878

663

12000

7,956,000

(120,000)

7,836,000

(2,350,800)

5485,200

240,000

______

5,725,200

3.6048

20,638,200.96

(4,000,000.00)

16,638,200.96

680

13500

9,180,000

(150,000)

9,030,000

(2,709,000)

6,321,000

240,000

______

6,561,000

3.6048

23651,092.8

(4,000,000.8)

19,651,092.8

In evaluating the project’s risk, we analyse the percentage change in NPV from the base scenario to

the worst case scenario and from the base case scenario to the best case scenario.

If the percentage change from the base case scenario to the worst case scenario is more than the

percentage change from the base case scenario to the best case scenario, then the project would be

highly risky.

Percentage change in NPV base case to worst case.

16,638,201−14,307,878

16,638,201×100= 14.01%

Percentage change in NPV base case to best case.

19,651,093−16,638,201

16,638,201×100 = 18.11%

Comment:

The project is less risky since percentage of worst is less than percentage of best.

Page 17

Illustration 2

A project with an initial cost of Sh. 2m and a nil salvage value is to be evaluated over its useful life of

4 years. The cost of capital applicable is 10% and the following information relates to it.

Variables Possible outcomes

Annual revenue (Sh.)

Variable costs (Sh.)

Fixed costs (Sh.)

Working capital at start

700,000

100,000

50,000

100,000

500,000

350,000

160,000

80,000

900,000

110,000

70,000

100,000

Tax rate is 40%

Required;-

Calculate the NPV under best case, base case and worst case scenario and comment on project risk.

Answer

Base case

scenario

Sh.

Worst case

scenario

Sh.

Best case

scenario

Sh.

Annual Revenue

Less Variable costs

Contribution

Less fixed cost

Before tax cash flows

Less tax @ 40%

After tax cash flows

Add: Depreciation tax shield

40% of Depreciation(40% × 500,000)

Net after tax cash flows

Discount factor

Present value of cash inflows

Less initial outlay

Investment in working capital

Net present value

700,000

(100,000)

600,000

(50,000)

550,000

(220,000)

330,000

200,000

530,000

3.1699

1,680,047

(2,000,000)

(100,000)

(419,953)

500,000

(350,000)

150,000

(160,000)

(10,000)

4,000

(6,000)

200,000

194,000

3.1699

614960.6

(2,000,000)

(80,000)

(1,465,039.4)

900,000

(110,000)

790,000

(70,000)

720,000

(288,000)

432,000

200,000

632,000

3.1699

2,003,376.8

(2,000,000)

(100,000)

(96,623.2)

Risk Evaluation

Percentage change in NPV base case to worst case

% worst = 𝑊𝑜𝑟𝑠𝑡 𝑁𝑃𝑉−𝐵𝑎𝑠𝑒 𝑁𝑃𝑉

𝐵𝑎𝑠𝑒 𝑁𝑃𝑉 x 100

(419953)− (1465039.4)

(419953)= 248.86%

Percentage change in NPV base case to best case

Ignore negatives in

% change

Page 18

% best case = 𝐵𝑒𝑠𝑡 𝑁𝑃𝑉−𝐵𝑎𝑠𝑒 𝑁𝑃𝑉

𝐵𝑎𝑠𝑒 𝑁𝑃𝑉

(419,953)− (96,623.2)

(419,953)= 77%

Comment:

The project is highly risky since percentage of worst is more than percentage of best.

3. Simulation Analysis

To simulate is to imitate. It involves conducting of a series of trial and error experiments. Where there

is a large number of random variables in an investment decision, simulation analysis may provide a

more satisfactory results in evaluating that project provided. Simulation can only apply when the

probability distribution of projects variables is given and a large number of trials are conducted to

reach a steady state.

Monte Carlo simulation and investment appraisal

Monte Carlo method

This section provides a brief outline of the Monte Carlo method in investment appraisal. The method

appeared in 1949 and is widely used in situations involving uncertainty. The method amounts to

adopting a particular probability distribution for the uncertain (random) variables that affect the NPV

and then using simulations to generate values of the random variables.

The basic idea is to generate through simulation thousands of values for the parameters or variables of

interest and use those variables to derive the NPV for each possible simulated outcome.

From the resulting values we can derive the distribution of the NPV.

Basic steps

1) Identify probabilistic variables

2) Determine cumulative probabilities

3) Assign RN – Ranges

NB: The Ranges will contain digits that correspond to the decimal places of probabilities i.e. if a

series has:

1) 1 decimal place probabilities that we use 1 digit (0 – 9)

2) 2 decimal places – 2 digits (00 – 99)

3) 3 decimal places – 2 digits (000 – 999)

Illustration 1

Determine random number range of the following distribution

Page 19

Price Probability

10

20

0.5

0.5

Solution

Price Cumm,ulative probabilities RN – Ranges

10

20

0.5

1.0

0 – 4

5 – 9

Illustration 2

X Probability

20

30

40

0.30

0.30

0.40

Solution

X Probability Cumulative

probability

RN – Range

20

30

40

0.30

0.30

0.40

0.30

0.60

1.00

00 – 29

30 – 59

60 – 99

Merits:

An increasingly popular tool of risk analysis, simulation offers certain advantages:

1) It facilitates the analysis and appraisal of highly complex, multivariate investment proposals with

the help of sophisticated computer packages.

2) It can cope up with both independence and dependence amongst variables. It forces decision-

makers to examine the relationship between variables.

Demerits:

1) Simulation is not always appropriate or feasible for risk evaluation.

2) The model requires accurate probability assessments of the key variables. For example, it may be

known that there is a correlation between sales price and volume sold, but specifying with

mathematical accuracy the nature of the relationship for model purposes may be difficult.

3) Constructing simulated financial models can be time-consuming, costly and requires specialized

skills, therefore. It is likely to be used to analyze very important, complex, and large-scale

projects.

4) It focuses on a project’s standalone risk. It ignores the impact of diversification, that is how a

project’s stand-alone risk will correlate with that of other projects within the firm and affects the

firm’s overall corporate risk.

Page 20

5) Simulation is inherently imprecise. It provides a rough approximation of the probability

distribution of net present value (or any other criterion of merit).

6) A realistic simulation model, likely to be complex, would most probably be constructed by a

management scientist, not the decision maker. The decision maker, lacking understanding of the

model, may not use it.

Illustration

XYZ Ltd intends to replace existing machines with a new one which is expected to increase its

profitability over the next 3 years.

Due to uncertainty in expected cash flows of this machine the following estimates with associated

probability has been provided.

Year Annual cash flows

Sh. 000

Probability

1

10,000

12,000

20,000

25,000

0.30

0.40

0.20

0.10

2

10,000

15,000

6,000

7,000

0.30

0.20

0.20

0.30

3 10,000

8,000

10,000

0.30

0.50

0.20

Cost of capital is 12% and the initial outlay of the machine shall be 27m.

Required;-

a) Using expected monetary value, calculate the expected NPV of investing in the machine.

b) Analyse the risk inherent in the situation above by simulating NPV calculation and hence

calculate the resultant NPV.

c) What is the probability of the new machine generating negative results?

NB: Use the following random numbers

43, 23, 66, 76, 24, 78, 90, 07, 45, 28, 46, 30, 19, 72, 83, 58, 49, 02

Solution

a) Expected NPV = Expected cash flows x Discount factor

Expected cash flows = ∑cash flows x Probability

Page 21

Sh.000

Year 1 =

Year 2 =

Year 3 =

14,300 (W1)

9,300

9,000

W1 : Expected cashflow ( Year 1)

10,000 x 0.3 + 12,000 x 0.4 + 20,000 x 0.20 + 25,000 x 0.1

NPV = 14,300 × PVIF12%1yr + PVIF12%2yr × 9,300 + 9,000×PVIFA x 12%3yr – 27m

14,300 × 0.8929 + 9,300 x 0.7972 + 9,000 x 0.7118 – 27,000,000

12,768.47 + 7,413.96 + 6,406.2 – 27,000 = Sh. (411.37) = -411.37

b) Random number ranges

Year cash flows

Sh.000

Probability Cumulative Probability RN

1

2

3

10,000

12,000

20,000

25,000

10,000

15,000

6,000

7,000

10,000

8,000

10,000

0.30

0.40

0.20

0.10

0.30

0.20

0.20

0.30

0.30

0.50

0.20

0.30

0.70

0.90

1.00

0.30

0.50

0.70

1.00

0.30

0.80

1.00

00-29

30-69

70-89

90-99

00-29

30-49

50-69

70-99

00-29

30-79

80-99

Simulation Work Sheet

No. Year 1

Discount factor =

0.8929

Year 2

Discount factor <

0.7972

Year 3

Discount factor ≤ 0.7118

RN Cash flows

Sh.000

RN Cash flows

Sh.000

RN cash flows

Sh. 000

Net present

value

1

2

3

4

5

6

43

76

90

28

19

58

12000

20000

25000

10000

10000

12000

23

24

07

46

72

49

10,000

10,000

10,000

15,000

7,000

15,000

66

78

45

30

83

02

Resultant

8,000

8,000

8,000

8,000

10,000

10,000

(2,618.8) (W1)

4,524.4 (W2)

8,988.9

(418.6)

(5,372.6)

2,790.8

7,894.1

Page 22

NPV

W1: 12,000 × 0.8929 + 100,000 × 0.7972 + 8,000 × 0.7118 – 27,000 = 2618.8

W2: 20,000 × 0.8929 + 10,000×0.7972 + 8,000×0.7118 – 27,000 = 4524.4

c) Probability of negative NPVS = 𝑁𝑜.𝑜𝑓𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑁𝑃𝑉𝑆

𝑁𝑜.𝑜𝑓𝑅𝑢𝑛𝑠 =

3

6 = 0.5

Illustration

The following probability estimates relates to a proposed project

Year Probability Cashflows

Sh.

Cost of equipment

Annual Revenue

Annual Running Costs

0

1-5

1-5

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

Cost of capital is 12%

Required;-

Assess how simulation method can be used to assess the above projects NPV.

Random numbers: 378420015689

Solution

Random number Ranges

Annual Revenue

Cashflows

Probability Cumulative RN

40,000

50,000

55,000

60,000

0.15

0.40

0.30

0.15

0.15

0.55

0.85

1.00

00-14

15-54

55-84

85-99

Annual costs cash flows

Year 1-5 25,000

30,000

35,000

0.10

0.25

0.35

0.10

0.35

0.70

00-09

10-34

35-69

Page 23

40,000 0.30 1.00 70-99

Simulation worksheet

Annual Revenue Annual costs

No. RN Cash flows

Sh.

RN Cash flows

Sh.

NPV

1

2

3

37

20

56

50,000

50,000

55,000

84

01

89

40,000

25,000

40,000

(3,952) (W1)

50,120

14,072

60,240

Therefore average NPV = 60,240

3 = 20,080

W1: NPV = 50,000 × PVIFA12%5yrs – 40,000 × PVIFA12%5yrs – 40,000

50,000×3.6048 – 40,000 × 3.6048 – 40,000 = Sh. (3,952)

4. Decision Theory Model

This model is well demonstrated by means of a decision tree

Decision Tree

This refers to a diagrammatic representation of the decision making process which indicates the

following

Decision alternatives

These are represented by the node (box node)

States of nature and associated probabilities

These are represented by the node (circle node)

Conditional payoffs

For each combination of the decision alternative and state of nature there is a payoff associated with

that combination. This may either involve a cash inflow or a cash outflow.

Decision tree is used to illustrate the process of decision making from the beginning of the project to

expiration i.e. from beginning of year 1 all through to the end of the project.

Process involved in Decision Making.

- Define the investment decision problem for example when entering a new market, expanding

existing projects etc.

- State the possible decision alternatives for example to build a new plant, lease the machine, buy

the machine that is small, medium or large.

- Identify all possible states of nature likely to affect decision outcome e.g. high demand or low

demand, boom or recession etc.

Page 24

- Estimate the probabilities associated with different states of nature identified above.

- Estimate conditional payoff for each combination of decision alternative and states of nature.

- Draw the decision tree.

- Calculate the expected value/expected monetary value of each of the alternatives available.

- Calculate the NPV of each of the available options we use the rollback technique in calculating

the resultant NPV of the project at hand i.e. we start from the furthest year then calculate NPV in

backward scenario up to year 0.

Merits

The sensitivity analysis has the following advantages:

It compels the decision maker to identify the variables affecting the cash flow forecasts

which helps in understanding the investment project in totality.

It identifies the critical variables for which special actions can be taken.

It guides the decision maker to concentrate on relevant variables for the project.

Demerits:

The sensitivity analysis suffers from following limitations:

The range of values suggested by the technique may not be consistent. The terms

‘optimistic’ and ‘pessimistic’ could mean different things to different people.

It fails to focus on the interrelationship between variables. The study of variability of one

factor at a time, keeping other variables constant may not much sense. For example, sales

volume may be related to price and cost. One cannot study the effect of change in price

keeping quantity constant.

Illustration 1

Researchers at Annex Electrical Ltd have invented a new television model. The company is ready for

pilot production and test marketing which will take one month at a cost of sh.40 million. It is expected

that there is a 70% chance of pilot production and test marketing being successful. In case of success,

Annex Electrical Ltd will build a plant at a cost of sh.300 million.

The plant will generate an annual cash flow of sh.60 million for 20 years if demand is high or an

annual cash flow of sh.40 million if demand is low. A high demand has a probability of 0.6. the

company’s required rate of return is 12%.

Required:

Advise the management of Annex Electrical Ltd on the best course of action.

Solution 60m

Annex Electrical Ltd (300m) high

0.6

0.7

0.4

(40m)

Successful low dmd

Page 25

40m

Do pilot

Unsuccessful 0.3

no pilot

ignore 0

Expected NPV when the plant is built

NPV = PV of Cash Inflows – Initial Outlay x PVIF12%,1 – Survey cost

Note 1: Expected cash flows =60,000,000 x 0.6 + 40,000,000 x 0.4

= Sh. 52,000,000 – This is annuity receivable for 20 years.

NPV successful = 52,000,000 x PVIFA12%20yrs – 300,000,000

52,000,000 x 7.4694 – 300,000,000 x 0.8929

= Sh. 120,538,800

NPV unsuccessful = 0

Node 2: NPV of Pilot testing

Sh. 120,538,800 x 0.7 + 0 x 0.3 – 40,000,000

= Sh. 44,377,160

The management should carry out the testing since if successful it would increase the value of the

firm i.e. it generates a positive NPV.

HINT: We are discounting the initial outlay of Kshs.300,000,000 since it will be paid at the end of

the first year.

Illustration II

ABC Ltd is a company operating in the telecommunications industry. The company intends to invest

in an equipment that would facilitate wireless internet connectivity to small and medium-sized

businesses. The equipment would cost sh.125 million.

Additional information:

1. Given the rapid technological change in the telecommunications industry, the equipment is

estimated to have a useful life of only three years with no salvage value.

2. The expected annual cash inflows from the project and their probabilities of occurrence are

dependent on the state of demand as shown below:

PVIFA 12%, 20 = 7.4694

PVIF 12%, 1 = 0.8929

Page 26

State of demand Probability Annual cash inflows (Sh.)

High

Average

Low

0.25

0.50

0.25

82.5 million

62.5 million

12.5 million

3. The company intends to purchase the equipment on 1 January 2019. However, the company has

the option of delaying the purchase to 1 January 2020 in order to obtain further information on the

project. The cost of the equipment, the cash inflows and their probabilities of occurrence are

expected to remain the same regardless of the project implementation date.

4. If the project is delayed to 1 January 2020 the cash inflows associated with each state of demand

will be known beforehand and the management would only purchase the equipment if a positive

net present value is expected.

5. The cost of capital is 12%.

Required:

(i)Using decision tree analysis, calculate the expected net present value (ENPV) standard deviation

and co-efficient of variation of the project as at 1 January 2019 under each of the two possible

implementation dates.

(ii)Advise the company on whether to invest in the equipment, and if so, on which date.

Solution

(i) Discount factor = PVIFA12%,3yrs = 2.4018

NPV (Invest today)

Node 1: 82.5 x PVIFA 12%,3 – Initial outlay

82.5 (1)

62.5 (2)

12.5 (3)

82.5 (4)

62.5 (5)

12.5 (6)

Invest today

Delay and invest after 1 year

Page 27

Sh. (million)

82.5 x 2.4018 – 125 = 73.1485

Node 2: 62.5 x 2.4018 – 125 = 25.1125

Node 3: 12.5 x 2.4018 – 125 = -94.9775

NPV (Delay and invest after 1 year)

Discount factor when delayed

PYIFA12% 4 years – PVIFA 12%, 1

3.0373 – 0.8929 = 2.1444

Node 4: 82.5 x 2.1444 – 125 x PVIF12%, 1

Sh. (millions)

82.5 x 2.1444 – 125x 0.8929 = 65.3005

Node 5: 62.5 x 2.1444 – 125 x 0.8929 = 22.4125

Node 6: 12.5 x 2.1444 – 125 x 0.8929 = -84.8075

Discount factor when delayed

PVIFA12%4yrs – PVIFA12%1yrs

3.0373 – 0.8929 = 2.1444

Investment of 1/1/2009

Expected NPV = ∑NPV x Probability

73,148,500 × 0.25 + 25,112,500 × 0.5 – 94,977,500 ×0.25 = Sh. 7,099,000

Standard deviation = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

Variance = ∑ returns x Probability

(73, 148,500– 7, 099,000)2× 0.25

(25, 112,500– 7, 099,000)2× 0.50

1.0906 × 1015

1.6224 × 1014

Page 28

(-94, 977,500– 7, 099,000)2×0.25

Variance

Standard deviation = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑠

2.6049 × 1015

3.85774 ×1015

= 62,110,707.61

Co-efficient variation = 62,110,707.61

7,099,000= 8.7492

Investment in 1/1/2010

Expected NPV = 65,300,500×0.25 + 22,412,500×0.5 – 84,807,500 × 0.25 = Sh. 6,329,500

Standard deviation

(65,300,500 – 6,329,500)2× 0.25 =

(22,412,500 – 6,329,500)2× 0.25 =

(-84,807,500 – 6,329,500)2× 0.25 =

Variance

Standard deviation = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑠

8.6939 × 1014

1.2933 × 1014

2.0765 × 1015

3.07522 × 1015

55,454,666.17

Coefficient of Variation = 8.7613

(ii) The company should invest in the project since NPV is positive.

The machine should be bought immediately rather than being delayed due to a lower risk as shown by

the coefficient of variation and also because of a higher expected return.

5. Certainty Equivalent

Under this model, risky cash flows are converted into riskless cash flows using a certainty equivalent

coefficient.

The riskless cash flows are then discounted using the Risk Free Rate of Return as the discount factor.

NPV = Present Value of Cash flows – Present Value of Cash outflows

PVCIF – PVCOF

Present value of Cash flows = (Expected Cash flows x Certainty Factor)

x Risk Free Discount Factor

Certainty Coefficient at the end of:

Year 1 = certain amount

Expected amount

Year 2 = (Certainty factor of year 1)2

Year 3 = (Certainty factor of year 1)3

Year n = (Certainty factor of year 1) n

Illustration 1

CV = 𝜎

𝐸𝑅

Page 29

Time Cash flow Certainty equivalent coefficient

0

1

2

3

(20)

10

12

15

1

0.85

0.90

0.92

Additional information

Cost of capital is 10%

Required:

Calculate NPV using certainty equivalent method.

Solution

First determine certainty cash flows by multiplying the given cashflows with the certainty factors.

Time Cashflows Certainty equivalent

coefficients

Certainty cashflows PVIF @10% PV

0

1

2

3

(20)

10

12

15

1

0.85

0.90

0.92

-20

8.5

10.8

13.8

1

0.9091

0.8264

0.7513

-20

7.73

8.93

10.37

NPV = 7.03

Illustration 2

A machine with an initial cost of Sh. 5,000,000 is expected to generate annual cash flows of Sh. 2.5m

over its economic life of 4 years. The company is indifference between a certain sum of Sh. 1815000

today and expected sum of Sh. 2.5m at the end of year 1. The risk free rate of interest is 7%.

Required;

Calculate the NPV of the project and advice the company on whether to implement the project.

Solution

Certainty factors at the end of:

Year 1 = 𝐶𝑒𝑟𝑡𝑎𝑖𝑛 𝑎𝑚𝑜𝑢𝑛𝑡

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤 =

1,815,000

2,500,000 = 0.726

Year 1 = 0.7260

Year 2 = (Certainty factor yr 1)2 = 0.72602 = 0.5271

Year 3 = 0.72603 = 0.3827

Year 4 = 0.72604 = 0.2778

NPV

Year Riskless Cash flows

Sh.

Discount factor

7%

Present value

Sh.

1 2,500,000 × 0.726 = 1,815,000 0.9346 1,696,299

Page 30

2

3

4

2,500,000 × 0.5271 = 1,317,750

2,500,000 × 0.3827 = 956,750

2,500,000 × 0.2778 = 694,500

0.8734

0.8163

0.7629

1,150,922.85

780,995.025

529,834.05

Present value of cash inflows

Less initial/Outlay

NPV

4,158,050.925

(5,000,000,000)

(841,949.075)

Illustration 3

David Majimbo is evaluating a project with a one year life and expected cash flow of sh. 5,000,000

receivable at year end. Shareholders require a return of 12%. The risk free rate is 6%.

Required:

Certainty equivalent coefficient. Interpret your result.

Solution

Present value of cash inflows = cash flows x CEC x Risk Free Discount Factor.

Present Value of cash inflows = 5,000,000 × PVIF12%1yr

5,000,000 × 0.8929 = Sh. 4,464,500

Expected Cash flows = 5,000,000

Risk Free Rate of Return = 6%

4,464,500 = [5,000,000x] ×PVIF6%1yr x CEC

4,464,500 = 5,000,000x × 0.9434 x CEC

Therefore CEC = 0.9465

Certainty Equivalent Coefficient = 0.9465

The management is at indifference whether to receive an uncertain amount of Sh. 5,000,000 after one

year or to receive (Sh. 0.9465 of 5,000,000) = Sh. 4.732500 today.

Merits of certainty equivalent

1) It is simple to calculate.

2) It is conceptually superior to time-adjusted discount rate approach because it incorporates risk by

modifying the cash flows which are subject to risk.

Demerits of certainty equivalent

1) This method explicitly recognizes risk, but the procedure for reducing the forecast of cash flows is

implicit and likely to be inconsistent from one investment to another.

2) The forecaster expecting reduction that will be made in his forecast, may inflate them in

anticipation. This will no longer give forecasts according to “best estimate”.

Page 31

3) If forecast have to pass through several layers of management, the effect may be to greatly

exaggerate the original forecast or to make it ultra conservative.

4) By focusing explicit attention only on the gloomy outcomes, chances are increased for passing by

some good investments.

6. Risk Adjusted Discount Rates (RADR)

Under this model, the discount factors for different projects are determined which are used in

calculating the NPV of the specific projects based on their risk level.

Traditionally, the same discount factor fact is used when evaluating different projects irrespective of

the difference in their level of risks.

Different projects are always affected by different factors which may not be similar to the risks of the

company hence the need to use Risk Adjusted Discount Factors/Discount factors when evaluating the

risks.

Decision Rule:

The risk adjusted approach can be used for both NPV and IRR.

If NPV method is used for evaluation, the NPV would be calculated using risk adjusted rate. If

NPV is positive, the proposal would qualify for acceptance, if it is negative, the proposal would

be rejected.

In case of IRR, the IRR would be compared with the risk adjusted required rate of return. If the

‘IRR’ exceeds risk adjusted rate, the proposal would be accepted, otherwise not.

Merits of risk adjusted discount rates

1) It is simple to calculate and easy to understand.

2) It has a great deal of intuitive appeal for risk-averse businessman.

3) It incorporates an attitude towards uncertainty.

Demerits of risk adjusted discount rates

1) The determination of appropriate discount rates keeping in view the differing degrees of risk is

arbitrary and does not give objective results.

2) Conceptually this method is incorrect since it adjusts the required rate of return. As a matter fact it

is the future cash flows which are subject to risk.

3) This method results in compounding of risk over time, thus it assumes that risk necessarily

increases with time which may not be correct in all cases.

4) The method presumes that investors are averse to risk, which is true in most cases. However,

there are risk seeker investors and are prepared to pay premium for taking risk and for them

discount rate should be reduced rather than increased with increase in risk.

Thus, this approach can be best described as a crude method of incorporating risk into capital

budgeting.

Illustration

Page 32

The finance manager of Biashara Ltd has suggested that the three projects in (b) above should be

analysed using the risk adjusted discount rate (RADR). The finance manager has developed the

following model to calculate the RADR for each project:

RADRj = Rf + RIj (K0 – Rf)

Where:

RADRj = Risk adjusted discount rate for project j.

Rf = Risk free rate.

RIj = Risk index for project j.

K0 = Cost of capital for the company.

Required:

(i) The risk adjusted discount rate for each project.

(ii) NPV for each project using the risk adjusted discount rates computed in (c) (i) above.

(iii) Based on the NPVs determined in (b) (ii) and (c) (ii) above, advise the company on which

project to pursue. Justify your answer.

Solution

Biashara Ltd

Cost of Capital = RF + (RM – RF) Beta

10% + (12% - 10%) 2.5

= 15%

Where RF – Risk Free Rate of Return from marketable securities, treasury bills/treasury bonds.

RM = Expected Market Return

β = Beta factor

Cash flows

Year Discount factor

15%

Project X

Sh. 000

Project Y

Sh. 000

Project Z

Sh. 000

0

1

2

3

4

1.000

0.8697

0.7561

0.6575

0.5718

(15,000)

6,000

6,000

6,000

6,000

(11000)

6000

4000

5000

2000

(19000)

4000

6000

8000

12000

NPV 2,130 1,673.7 1,137

Project X

NPV = 6,000 x PVIFA4yrs15% - 15000

= 6000 x 2.8560 – 15000

= Sh. 2130

RADRj = Rf + RIj (Ko – Rf)

Project X - 10% + 1.8 (15% - 10%) = 19%

Y - 10% + 1.0 (15% - 10%) = 15%

Page 33

Z - 10% + 0.6 (15% - 10%) = 13%

Project X Using RADR

Year Cash flows

Sh. 000

0

1

2

3

4

(15,000)

6,000

6,000

6,000

6,000

NPV=6000 x PVIFA19%4yrs – 15000

=831.6

Project Y

Year Cash flows

Sh. 000

Discount factor

(15%)

Present Value

Sh. 000

0

1

2

3

4

(11,000)

6,000

4,000

5,000

2,000

1.0000

0.8696

0.7561

0.6575

0.5718

NPV

(11,000)

5,217.6

3,024.4

3,287.5

1,143.6

1,673.1

Project Z

Year Cash flows

Sh. 000

Discount factor

(15%)

Present Value

Sh. 000

0

1

2

3

4

(19,000)

4,000

6,000

8,000

12,000

1.0000

0.8850

0.7831

0.6931

0.6133

NPV

(19,000)

3540

4,698.6

5,544.8

7,359.6

2,143

The company should invest in project Z since even after incorporating Risk specific to each project, it

offers the highest return.

7. Utility Theory (Curves)

Utility refers to the satisfaction derived by an individual based on the amount of wealth purchased.

This theory explains how an investor will react if present with different alternatives that are risky.

If an investor is willing to pay more than his expected values, such an investor is known as Risk Taker

Investor.

Page 34

If an investor is willing to pay less than his expected return value, such an investor is known as

Risk Averse Investor.

In case the investor is willing to pay the same amount as his expected return, such an investor is

known as a Risk Neutral Investor.

As regards the attitude of individual investors towards risk, they can be classified in three categories. ·

Risk-averse investors attach lower utility to increasing wealth i.e. for a given wealth or return, they

prefer less risk to more risk. ·

Risk-neutral investors attach same utility to increasing or decreasing wealth i.e. they are indifferent to

less or more risk for a given wealth or return.

Risk-seeking investors attach more utility to the potential of additional wealth to the loss from the

possible loss from the decrease in wealth. I.e. for earning a given wealth or return, they are prepared

to assume higher risk. It is well established by many empirical studies that individuals are generally

risk averse and demonstrate a decreasing marginal utility for money function.

1.3 INCORPORATING CAPITAL RATIONING IN CAPITAL INVESTMENT

APPRAISAL

Capital rationing implies investment in projects within limited capital resources. It is the process of

allocating money among different projects, where the amount of money to be invested is limited.

Companies ration their capital and investments among different opportunities as countries use

rationing of food. In case of capital rationing, the company may not be able to invest in all profitable

projects.

Types of capital rationing

1. Internal/Soft Capital Rationing

In this case the decisions of the management leads to the company not being able to raise all the

required funds necessary for undertaking all profitable projects examples.

(i)Issue of additional shares to the public may be avoided so as not to dilute the firm’s earnings per

share.

(ii)Issue of additional debt may be avoided so as not to increase the firm’s gearing/financial risk.

(iii)Use of debt finance may be avoided so as not to increase the fixed finance costs inform of

interests.

(iv)The management may opt to maintain their investments at a level that can only be financed by

internally generated funds.

2. External/Hard capital rationing

It arises due to external factors which are beyond the control of the management, i.e. the firm is

unable to raise all the required funds due to the restrictions from the financial markets among other

external factors for example;

i) Raising of funds from the public through ordinary shares or debentures is not possible for a

company that is not listed (Quoted at the securities market).

ii) Raising of funds through debt may be difficult for a firm without collateral to pledge a security.

iii) The existing shareholders/investors may be unwilling to increase their investments to the firm.

Page 35

iv) The existing contractual obligations may prohibit the firm from raising additional funds from

other sources.

v) Government policies with respect to regulations of financial markets may prevent the firm from

additional borrowing through an increase in minimum lending rates.

Profitability Index = 𝑃𝑉𝑜𝑓𝑐𝑎𝑠ℎ 𝑖𝑛 𝑓𝑙𝑜𝑤𝑠

𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡 𝑜𝑢𝑡𝑙𝑎𝑦

Single period capital rationing (one period capital rationing)

It is where the limitation of fund is not expected to extend beyond the current financial year.

This is applicable when limits are placed on the availability of finance for positive NPV for one year

only and capital is freely available in all the rest of periods.

There are some additional assumptions in single period rationing which are very important to consider

here which include:

(i) If a firm does not undertake a project `now'the period of capital scarcity, the opportunity is lost

in other words, the project cannot be deferred until the capital is available.

(ii) The outcome of each project is known with certainty so that the choice between the projects is not

affected by considerations of risk.

(iii) The projects are divisible. This means that we can undertake 50% of project A and 50% of project B.

The basic approach will be to rank the projects in such in such a way that NPV can be maximised

from the use of available finances using profitability indices

Ranking the projects using NPV will be incorrect in this scenario because NPV basis will lead to

select the ‘big’ projects, each of which has a high individual NPV but which have a lower NPV than a

large number of smaller projects with lower individual NPVs.

Therefore, ranking should be made in terms of Profitability index

Illustration 1

Kihingo limited is considering five project proposals as summarised below;

Project Initial cost sh.

‘million’

Annual Rev. Sh.

‘million’

Annual fixed cost sh.

‘million’

Life of the project

(Years)

A

B

C

D

E

10

30

15

12

18

20

30

18

17

8

5

10

6

8

2

3

5

4

10

15

Additional information:

1. The variable costs is 40% of the annual revenue

2. Projects D & E are mutually exclusive.

3. Each project can only be undertaken once and each is divisible.

Page 36

Assume that;

All the cash flows are confined to within the life of each project

The cost of capital is 10%

No inflation exists

There is no risk

No taxes exist

All cash flows occur on anniversary dates.

Required:

Assuming that the company has a limit of sh.40m for investment in projects at time 0 (zero),

determine the optimal allocation of the sh.40 million among the projects and the resultant maximum

net present value (NPV) obtained.

Solution

Kihingo Ltd

We evaluate the profitability of each project using profitability index.

PI = 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓𝑐𝑎𝑠ℎ 𝑖𝑛 𝑓𝑙𝑜𝑤𝑠

𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦

Project Initial

outlay

Sh. ‘m’

Contribution

Sh. ‘m’

Fixed

costs

Sh. ‘m’

Cashflows

Sh. ‘m’

Period

Yrs

DF

10%

PV PI

A

B

C

D

E

10

30

15

12

18

12

18

10.8

10.2

4.8

5

10

6

8

2

7

8

4.8

2.2

2.8

3

5

4

10

15

2.4869

3.7908

3.1699

6.1446

7.6061

17.4083

30.3264

15.2155

13.5181

21.2970

1.74083

1.01088

1.014368

1.12651

1.183171

Optimal Allocation of Sh. 40m available

Project Initial Outlay

Sh. m

PV of Cash Inflows

Sh. m

NPV

Sh. m

A

E

C

10

18

12

40

17.4083

21.29708

12.172416

Total NPV

7.4083

3.29708

0.172416

10.877796

NOTE: Optimality is obtained through ranking the projects using profitability index. The higher the

better the project.

PV of C = 12

15 x 15.2155 = 12.1724

Illustration 2

Robin, a multi-product company, is considering four investment projects, details of which are given

below. Development costs already incurred on the projects are as follows;-

Page 37

A

Sh.

100,000

B

Sh.

75,000

C

Sh.

80,000

D

Sh.

60,000

Each project will require an immediate outlay on plant and machinery, the cost of which is estimated

as follows;-

A

Sh.

2,100,000

B

Sh.

1,400,000

C

Sh.

2,400,000

D

Sh.

600,000

In all four cases the plant and machinery has a useful life of five years at the end of which it will be

valueless.

Unit sales per annum for each project, are expected to be as follows:

A

150,000

B

75,000

C

80,000

D

120,000

Selling price and variable costs per unit for each project are estimated below:

Selling price

Materials

Labour

Variable overheads

A

Sh.

30.00

7.60

9.80

6.00

B

Sh.

40.00

12.00

12.00

7.00

C

Sh.

25.00

4.50

5.00

2.50

D

Sh.

50.00

25.00

10.00

10.50

The company charges depreciation on plant and machinery on a straight line basis over the useful life

of the plant and machinery. Development costs of projects are written off in the year that they are

incurred. The company apportions general administration costs to projects at a rate of 5% of selling

price. None of the above projects will lead to any actual increase in the company’s administration

costs.

Working capital requirements for each project will amount to 20% of the expected annual sales value.

In each case this investment will be made immediately and will be recovered in full when the projects

end in five years’ time.

Funds available for investment are limited to sh.5,200,000. The company’s cost of capital is estimated

to be 18%.

Required:

(a) Calculate the NPV of each project

(b) Calculate the profitability index for each project and advise the company which of the new

projects, if any, to undertake. You may assume that each of the projects can be undertaken on a

reduced scale for a proportionate reduction in cash flows. Your advise should state clearly your

order of preference for the four projects, what proportion you would take of any project that is

scaled down, and the total NPV generated by your choice.

(c) Discuss the limitations of the profitability index as a means of dealing with capital rationing

problems.

Page 38

Solution

(a) The first step is to calculate the annual contribution from each project, together with the working

capital cash flows. These cash flows, together with the initial outlay, can then be discounted at the

cost of capital to arrive at the NPV of each project. Development costs already incurred are

irrelevant. There are no additional administration costs associated with the projects and

depreciation is also irrelevant tax rate is not provided.

First, calculate annual contribution

Unit sales

Selling price per unit

Material cost per unit

Labour cost per unit

Variable overheads per unit

Sales per annum

Materials

Labour

Variable overheads

Annual contribution

Working capital requirement

(20% annual sales value)

A

150,000

Sh.

30.00

7.60

9.80

6.00

Sh.000

4,500

(1,140)

(1,470)

(900)

990

A

Sh.000

900

B

75,000

Sh.

40.00

12.00

12.00

7.00

Sh.000

3,000

(900)

(900)

(525)

675

B

Sh.000

600

C

80,000

Sh.

25.00

4.50

5.00

2.50

Sh.000

2,000

(360)

(400)

(200)

1,040

C

Sh.000

400

D

120,000

Sh.

50.00

25.00

10.00

10.50

Sh.000

6,000

(3,000)

(1,200)

(1,260)

540

D

Sh.000

1,200

Project A

Example:

Time 0 = Initial outlay + working capital

= 2100 + 900 = 3000

1 – 5 990

5 900 (Release of working capital)

Time

0

1 – 5

5

NPV

A

(3,000)

990

900

489

B

(2,000)

675

600

373

C

(2,800)

1040

400

627

D

(1,800)

540

1200

413

PVIF18%

1

3.1272 (w1)

0.4371

Working 1 (W1)

NPV for A = -3000 × 1 + 990 × 3.1272 + 900 × 0.4371 = 489.318

(b) The probability index provides a means of optimizing the NPV when there are more projects

available which yield a positive NPV than funds to invest in them. The profitability index

measures the ratio of the present value of cash inflows to the initial outlay and represents the net

present value per sh.1 invested.

Page 39

(c)

Project

A

B

C

D

PV of inflows

Sh.000

3,489

2,373

3,427

2,213

Initial outlay

Sh.000

3,000

2,000

2,800

1,800

Ratio

1.163

1.187

1.224

1.229

Ranking

4

3

2

1

Project D has the highest PI ranking and is therefore the first choice for investment.

Amount available

Less investment D

Less investment C

Invest all in B

5,200

(1,800)

3,400

(2800)

600

(600)

0

NPV

413

627

112(w1)

1152

W1 = 600

2000× 373 = 112

Therefore total NPV = 1152

(d) The probability index (PI) approach can be applied only if the projects under consideration fulfill

certain criteria, as follows:

(i) There is only one constraint on investment, in this case capital. The PI ensures that

maximum return per unit of scarce resource (capital) is obtained.

(ii) Each investment can be accepted or rejected in its entirety or alternatively accepted on a

partial basis.

(iii) The NPV generated by a given project is directly proportional to the percentage of the

investment undertaken.

(iv) Each investment can only be made once and not repeated.

(v) The company’s aim is to maximize overall NPV.

If additional funds are available but at a higher cost, then the simple PI approach cannot be used

since it is not possible to calculate unambiguous individual NPVs.

If certain of the projects that may be undertaken are mutually exclusive then sub-problems must

be defined and calculations made for different combinations of projects. This can become a very

lengthy process. These assumptions place limitations on the use of the ration approach. It is not

appropriate to multi-constraint situations when linear programming techniques must be used.

Each project must be infinitely divisible and the company must accept that it may need to

undertake a small proportion of a given project. This is frequently not possible in practice. It is

also very unlikely that there is a simple linear relationship between the NPV and the proportion

of the project undertaken; it is much more likely that there will be discontinuities in returns.

Possibly a more serious constraint is the assumption that the company’s only concern is to

maximize NPV. It is possible that there may be long-term strategic reasons which mean that an

investment with a lower NPV should be undertaken instead of one with a higher NPV, and the

ratio approach takes no account of the relative degrees of risk associated with making the

different investments.

Page 40

Illustration 3

Emalex Ltd has a budget of sh.240 million for investment in various projects. The finance manager

has presented the following proposals for immediate investment. The first cash return is expected in

12 months and at annual intervals thereafter.

Project 2012

Sh.

“million

”

2012

Sh.

“million

”

2012

Sh.

“million

”

2012

Sh.

“million

”

2012

Sh.

“million

”

2012

Sh.

“million

”

2012

Sh.

“million

”

Nwt present

value

(NPV)

Sh.

“million”

Internal

rate of

return

(IRR)

%

A

B

C

D

E

F

(124)

(128)

(48)

(200)

(24)

(80)

56

16

26

60

5

49

20

24

24

100

11

50

24

40

12

50

15

-

-

42

2

58

42

-

-

84

-

-

-

-

-

(16)

-

-

-

-

11

22.2

4

14.4

3.8

5.8

16

13

15

13

17

15

There is no option to delay any of the projects. All projects except project A can be scaled down but

cannot be scaled up. The company has a current cost of finance of 10% but it would take one year to

establish further funding at that rate. Further funding for short periods could be arranged at a higher

interest rate.

Required:

(i)The projects that should be undertaken in the order if their priority

(ii)The net present value (NPV) and the internal rate of return (IRR) for the projects undertaken.

(iii)Estimate and advise on the maximum interest rate that the company should pay to finance all the

remaining projects available.

Solution

To determine the projects to undertake, we rank all the projects using profitability index.

Profitability Index PI = 𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑣𝑎𝑙𝑢𝑒𝑜𝑓𝑐𝑎𝑠ℎ𝑖𝑛𝑓𝑙𝑜𝑤𝑠

𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑜𝑢𝑡𝑙𝑎𝑦

Projects Initial outlay

Sh. ‘m’

PV of Cash Inflows

Sh. ‘m’

PI Rank

A

B

C

D

E

F

124

128

48

200

24

80

135

141.8

52

214.4

27.8

85.8

1.0887

1.1078

1.0833

1.072

1.1583

1.0725

3

2

4

6

1

5

Optimal Allocation

Project Initial Outlay

Sh. ‘m’

E 24

Page 41

B

A

Total

92

124

240

Since project A is not divisible we undertake the whole of it but only a portion of project B due to its

divisibility nature i.e. 92/128× 100 = 71.875% of project B is undertaken.

Net Present Value

Project Initial Outlay

Sh. m

NPV

Sh. m

E

B

A

24

92

124

3.8

9.92

11

24.72

NPV = (13.801756)

NPV @ 10% = 24.72

NPV 18% = (13.80)

IRR = Lower Rate +(NPV at LR) (Higher Rate – Lower Rate)

NPV at LR-NPV at HR

10 + (24.72 ) (18-10) = 15.13%

24.72--13.80

The maximum interest rate to be paid is the discount factor that gives a O NPV i.e. IRR

End of year

Sh. ‘m’

2012

Sh. ‘m’

2013

Sh. ‘m’

2014

Sh. ‘m’

2015

Sh. ‘m’

2016

Sh. ‘m’

2017

Sh. ‘m’

2018

Project E

B

A

Net c. flow

Discount factor

18%

0

(24)

(92)

(124)

(240)

1.0000

(240)

1

5

11.5

56

72.5

0.8475

61.44375

2

11

17.3

80

108.3

0.7182

77.78106

3

15

28.75

24

67.75

0.6086

41.23265

4

4.2

30.2

____

34.4

0.5158

17.74352

5

-

60.4

___

60.4

0.4371

26.40084

6

-

(4.31)

____

(4.31)

0.3704

1.596424

Page 42

Project Initial Outlay

Sh. ‘m’

NPV

Sh. ‘m’

B

C

D

F

36

48

200

80

NPV

3.88

4

14.4

5.8

28.08

NPV = (26.331855)

IRR =Lower Rate + (𝑁𝑃𝑉𝑎𝑡𝐿𝑅

𝑁𝑃𝑉𝑎𝑡𝐿𝑅−𝑁𝑃𝑉𝑎𝑡𝐻𝑅) (Higher Rate – Lower Rate)

10 + (28.08 ) (18-10) = 14.13%

28.08—26.33

Single period capital rationing and Indivisible Projects

These are projects which can only generate cash flow upon completion.

In calculating the optimal allocation/combination we use trial and error combination method, any

surplus funds shall be invested elsewhere to generate additional NPV.

Illustration

Independent projects are available for evaluation whose information is as follows.

Project Initial outlay PV of Cash inflows NPV

A

B

C

D

30M

45M

40M

60M

40M

55M

60M

100M

10M

10M

20M

40M

Additional Information

There is a capital limit of Sh. 95m and each project is indivisible. Extra amount can be invested at a

return of 12% to infinity and the cost of capital is 10%.

Required

Determine the optimal allocation of the above projects.

End of year

Sh. ‘m’

2012

Sh. ‘m’

2013

Sh. ‘m’

2014

Sh. ‘m’

2015

Sh.‘m’

2016

Sh. ‘m’

2017

Sh. ‘m’

2018

Project B

C

D

F

Net Cash flows

Discount factor @ 18%

0

(36)

(48)

(200)

(80)

(364)

1.0000

1

4.5

24

60

49

137.5

0.8475

2

6.75

24

100

50

180.75

0.7182

3

11.25

12

50

____

73.25

0.6086

4

11.81

2

58

____

71.81

0.5158

5

23.63

-

-

___

23.63

0.4371

6

(1.69)

-

-

____

(1.69)

0.3704

Page 43

Solution

Combination Initial outlay

of combination

Excess

funds

NPV of

combination

NPV of

excess

funds

Combined

NPV

A & B

A & C

A & D

B & C

75M

70M

90M

85M

20M

25M

5M

10M

20M

30M

50M

30M

4M

5M

1M

1M

24M

35M

51M

31M

Workings: Excess funds of NPV

A & B Cash inflows = 12% x 20,000,000 = 24M to infinity

Recall PVIFAr%∝ =1

𝑟

R = 10% = Cost of Capital

NPV = PVIFA10% ∝ - 20M = 2.4M × PVIFA10% ∝ - 20M = 2.4M ×1

0.1 – 20 = 4M

A & C = 12% × 25M ×1

0.1 – 25M = 5M

A & D = 12% ×5M ×1

0.1 – 5M = 1M

B & C = 12% ×10M ×1

0.1 – 10M = 1M

The optimal combination would be the combination of project A & B because it gives the highest

NPV.

Multi Period Capital Rationing

It is a situation where the period of capital rationing is expected to be more than 1 year (many period).

In this case it is impossible to rank the projects using NPV, Profitability Index or IRR.

A mathematical programming model such as linear programming shall be adopted in this case.

Illustration 1

A company intends to invest in two divisible projects A and B. Each project can be undertaken fully

or partially.

Details of the project are:

End of year Project A

Sh. 000

Project B

0

1

2

3

(10,000)

(20,000)

(30,000)

120,000

(20,000)

(10,000)

-

70,000

Page 44

Additional information

Cost of capital is 10% and no project can be postponed.

Available funds are restricted as follows:

Year Available funds

Sh. 000

0

1

2

25,000

30,000

20,000

Funds not utilised in year 1 will not be available in the subsequent years.

Required;-

Formulate the linear programming model to determine the optimal allocation of funds.

Solution

Step 1

Objective function /Maximise NPV

Project A Project B

Year Discount factor cash flows Present Value cash flows present value

0

1

2

3

1.0000

0.9091

0.8264

0.7513

(10,000)

(20,000)

(30,000)

100,000

(10,000)

(18,182)

(24,792)

75130

(20,000)

(10,000)

-

60,000

(20,000)

(9091)

-

45078

Net Present Value 22,156 15,987

Objective function: Maximise NPV = 22156A + 15987B

Subject to (Determine of decision constraints)

1. 10,000 A + 20,000 B≤20,000

2. 20,000 A + 10,000 B ≤ 25,000

3. 30,000 A ≤ 20,000

A,B≥0 (Non negativity function)

Conversion of constraints into equations for plotting on graph

1. A + 2B = 2

2. 2A + B = 2.5

3A = 2 therefore A = 2/3 = 0.67

Graph

B

When A = 0 B = 1

When B = 0 A = 2

When A = 0 B = 2.5

When B = 0 A = 1.25

Page 45

3.0

2.5

2.0

1.5

1.0

0.5 B C

A D

0.5 1.0 1.5 2.0 2.5 3.0

A

Optimal Allocation

Corners Objective function – Maximise NPV 22,156A + 22,156A + 15,987 B

A (0,0) therefore NPV = 22156 x 0 + 15987 x 0 = 0

B (0.67,0) therefore NPV = 22156 x 0.67 + 15,987 x 0 = 14,844.52

C(0.67,0.8) therefore NPV = 22156 x 0.67 + 15,987 x 0.8= 27,634.12

D (0,1.25) therefore NPV = 22156 x 0 + 15987 x 1.25 = 19,983.75

Conclusion

Of the total investment 0.67 should be made in project A and 0.8 in project B since the optimal

allocation as it generates the highest NPV.

Illustration 2

A company has 2 independent projects which are divisible and generate cashflows as follows:

Year 0 1 2 3

Project: x

Sh. (‘m’)

(10)

(22)

(30)

(15)

(35)

-

150

100

(Terminal cashflows)

The cost of capital is 12% and for each of the years 0,1 and 2 only sh. 24m, 30m and 28m is available

for investment purposes respectively.

Required;-

Determine the optimal project combination using LP model

Solution

Objective function

Project x Project y

Year Discount factor Cash flows Present Value cash flows Present Value

0 1.0000 (10) (10) (22) (22)

Page 46

1

2

3

0.8929

0.7972

0.7118

(30)

(35)

150

(26.787)

(27.902)

106.77

(15)

-

100

(13.3935)

-

71.18

Net Present Value 42.081 35.7865

Objective function maximise NPV = 42.081x + 35.7865y

Subject to (decision constraints)

10x + 22y ≤ 24

30x + 15y ≤ 30

35x ≤ 28

x, y ≥ 0

s

Conversion of constraints into equations

10x + 22y = 24

30x + 15y = 30

35x = 28 x = 0.8

Graph

0 0.5 1.0 1.5 2.0 2.5 3.0

3.0

2.5

2.0

1.5

1.0

0.5

0

B

C A D

10X + 22Y≤20

30X + 15Y ≤24

35X ≤2.8

When x = 0 y = 1.1

When y = 0 x = 2.4

When x = 0 y = 2

When y = 0 x = 1

Page 47

Optimal Allocation

Corners objective function maximise NPV (42.081x + 35.7865)

A (0,0) = 0

B (0,8.0) 0.8 x 42.081 + 0 x 35.7865 = 33.6648

C (0.6,0.8) 0.6 x 42.081 + 0.8 x 35.7865 = 53.8778

D (0,1.1) 0 x 42.081 + 1.1 x 35.7865 = 39.36515

Conclusion

Of the total investment 0.6m should be made in project x and 0.8m in project y since it is the optimal

allocation.

1.4 INCORPORATING INFLATION IN CAPITAL INVESTMENT APPRAISAL

INVESTMENT DECISION UNDER INFLATION

Inflation is the general increase in the price of goods and services.

Inflation can either be general or specific inflation.

General inflation affects the overall investors required rate of return.

Specific inflation only affects individual cash flows components.

Dealing with inflation in Net Present Value calculation two methods are used:

Real Method

Money Method/nominal

Real method

Under this method, cash flows are not inflated but the discount factor cost of capital /money rate is

adjusted to be the rate of return using the fisher formula

Using fisher formula (1 + r) (1 + i) = (1 + m) where

r = real rate of return

i = Inflation rate

m = money market rate of return /cost of capital

Make r the subject of the formulae

Page 48

Therefore r= 1+𝑚

1+𝑖 – 1 This rate only applies in case of general inflation.

Money Method/ Nominal

Under this method we inflate the real cash flows into money cash flows and we discount them using

the money rate of return/cost of capital that is the discount factor shall be applied without being

adjusted for inflation.

Illustration

Two mutually exclusive projects are available with the following information.

Project A

Its initial outlay is Sh. 10m, nil scrap value and economic life of 5 years.

Annual revenue expected is Sh. 6m and contribution 0.9 of total revenue

Depreciation is on straight line basis and general inflation affecting the project is 2%.

Project B

Its initial outlay is Sh. 10m, nil scrap value and economic life of 5 years.

Annual revenue is Sh. 7m and variable cost of Sh. 1.5m p.a. Selling price inflation is at 3% while

variable cost inflation is at 2%.

Required:

Advice the management of a given company on which project to implement assessing the tax rate of

20% cost of capital 12%.

NB: Project B would also require an investment in working capital as initial cost which would be

affected by general inflation rate of 2%, working capital of Sh. 300,000.

Solution

Project A

Since the project is affected by general inflation rate, we can either adopt real or money method.

1) Real method

r =1+𝑚

1+𝑖 – 1 =

1.12

1.02 – 1 = 0.0980 × 100 = 9.8% =10%

NPV=Present value of cash inflows - Initial Outlay

Initial outlay = sh. 10,000,000

Year 1– 5 years Annual after Tax Cash flows

Sh.

Revenue

Contribution (0.9 × 6m)

Less Tax

6,000,000

5,400,000

(1,080,000)

Page 49

Annual after tax cash flows

Add: depreciation tax shield

4,320,000

400,000

4,720,000

4720,000 × PVIFA5yrs10% - 10,000,000

4720000 × 3.7908 – 10,000,000

= Sh. 7,892,576

2) Money Method

Initial outlay = Sh. 10,000,000

Annual after Tax cash flows

Year 1

Sh.

2

Sh.

3

Sh.

4

Sh.

5

Sh.

Contribution

Inflation (1.02)n

Real contribution

Less Tax 20%

Add depreciation Tax shield

Discount factor 12%

5,400,000

1.02

5,508,000

(1,101,600)

4,406,400

400,000

4,806,400

0.8929

5,400,000

1.04

5,616,000

(1,123,200)

4,492,800

400,000

4,892,800

0.7972

5400000

1.06

5,724,000

(1,144,800)

4,579,200

400,000

4,979,200

0.7118

5,400,000

1.08

5,832,000

(1,166,400)

4,665,600

400,000

5,065,600

0.6355

5,400,000

1.10

5,940,000

(1,188,000)

4,752,000

400,000

5,152,000

0.5674

Present value of Cash inflows

Less initial Outlay

NPV

Sh.

17,878,802.88

10,000,000

7,878,802.88

Project B

Initial Outlay = 10,000,000 + 300,000 = 10,300,000

Annual after Tax cash flows

Year 1

Sh.

2

Sh.

3

Sh.

4

Sh.

5

Sh.

Revenue.7m (1.03)n

v.cost 1.5m (1.02)n

Contribution

Tax @ 20%

Add depreciation tax shield 20%

Less increase in w.c

7,210,000

(1,530,000)

5,680,000

(1,136,000)

4,544,000

400,000

4,744,000

(6000)

7,426,300

(1,560,600)

5,865,700

(1,173,140)

4,692,560

400,000

5,092,560

(6120)

7,649,089

(1,591,812)

6,057,277

(1,211,455)

4,845,822

400,000

5245822

(6242.4)

7,878,562

(1,623,648)

6,254,914

(1,250,983)

5,003,931

400,000

5,403,931

(6,367.248)

8,114,919

(1,656,121)

6,458,798

(1,291,760)

5,167,038

400,000

5,567,038

(6494.593)

Note: We use the inflation rate while

discounting the cash flow

Page 50

Net after tax Cash flows 4,938,000 5086440 5,239,579.6 5,397563.75 5,560,543.40

Terminal cash flows

Release of working capital

Salvage value

Sh.

331,224

Nil

331,224

Investment in working capital

Year 1

Sh.

2

Sh.

3

Sh.

4

Sh.

5

Sh.

Beginning balance

Inflation @ 2%

End balance

300,000

6,000

306,000

306,000

6,120

312,120

312,120

6,242.4

318,362.4

318,362.4

6,367.248

324,729.648

324,729.648

6,494.593

331,224.241

Net Present Value

End of year Cash flows

Sh.

Discount factor

12%

Present value

Sh.

1

2

3

4

5

5

4,944,000

5,092,560

4,845,822

5,003,931

5,167,038

331,224

0.8929

0.7972

0.7118

0.6355

0.5674

0.5674

4,409,140.2

4,059,788.822

3,449,256.1

3,179,998.151

2,931,777.361

187,936.4967

Present value of cash inflows

Less initial outlay

NPV

18,223,154.14

(10,300,000.00)

7,923,154.141

The management should implement project B as it generates the highest NPV.

1.5. EVALUATION OF PROJECTS OF UNEQUAL LIVES

Evaluation projects with different economic/useful lives.

When selecting investment projects that compete with each other that is mutually exclusive projects

and the same projects have different economic lives, it becomes difficult since a direct comparison

between these two projects would not provide a fair result for example project A has a useful life of 4

years and B with a useful life of 6 years, under normal circumstances B will provide the highest NPV

since it would generate more cash flows for two more years.

To recommend implementation of project B only based on a higher NPV may therefore not be a

sound reason.

This is because there may be a possibility of reinvesting cash flows associated with project A over the

remaining two years.

Page 51

This fact is more often ignored since NPV of the projects at the end of economic useful lives are

always compared.

In such circumstances where two projects are compared with different economic lives we can either

use;

- Equivalent annuity model

- Replacement chain analysis/constant scale replication model

Equivalent Annuity Model

Under this model the NPV of individual projects is divided by the present value interest factor of

annuity over its useful life using the cost of capital as the discount factor

i.e. Equivalent Annuity =𝑁𝑃𝑉

PVIFAr%nyrs

The period with the highest equivalent annuity NPV or the lowest equivalent annuity cost is

undertaken.

Replacement Chain Analysis/constant scale replication method

Under this model, the project with the shorter useful life is reinvested for a period equivalent to the

remaining useful life of the other project and the total NPV is calculated.

It assumes that the projects are directly comparable in that they can multiply each other in terms of

economic lives.

Illustration

As a newly appointed manager of Tena Ltd, you are required to choose between the following

mutually exclusive projects.

Net cash flows in millions of shillings

Year Project A Project B

0

1

2

3

4

5

(250)

200

500

(1,000)

300

400

600

500

The company’s cost of capital for project under similar risk levels is 12%.

Required;-

(i) The net present value (NPV) of each project using the constant scale finite period replication

criteria.

(ii)Annual equivalent value (AEV) of each project.

(iii) Make a decision on which project to undertake.

Solution

Page 52

Tena Ltd

Step 1

A

End of year Discount factor

12%

Cycle I

Sh. ‘m’

Cycle II

Sh. ‘m’

Total cash flows

Sh. ‘m’

Present value

Sh. ‘m’

0

1

2

3

4

1.0000

0.8929

0.7972

0.7118

0.6355

(250)

200

500

-

-

(250)

200

500

250

200

250

200

500

(250)

178.58

398.6

142.36

317.75

Net present value 587.99

B

End of year Discount factor Cashflows Present Value

0

1

2

3

4

1.0000

0.8929

0.7972

0.7118

0.6355

(1000)

300

400

600

500

Net present value

(1000)

267.87

318.88

427.08

317.75

331.58

Project A should be undertaken since it has the highest NPV

Annual Equivalent Value

End of year Discount factor

12%

Project A Project B

0

1

2

3

4

1.0000

0.8929

0.7972

0.7118

0.6355

Net present value

(250)

200

500

_____

327.18

(1000)

300

400

600

500

331.58

Equivalent Annuity = 𝑁𝑃𝑉

𝑃𝑉𝐼𝐹𝐴𝑟% 𝑛𝑦𝑟𝑠

A →327.18

𝑝𝑣𝑖𝑓𝑎 12% 2 𝑦𝑒𝑎𝑟𝑠

327.18

1.6901= 193.59

B →331.58

𝑝𝑣𝑖𝑓𝑎 12% 4 𝑦𝑒𝑎𝑟𝑠

Page 53

331.58

3.0373= 109.17

Project A should be implemented since it has the highest equivalent annuity.

Replacement Analysis

At times the company is faced with the problem of the right moment of either replace a project with

another one or to abandon it all together.

Under this method, the decisions are made on whether to replace a project with another or not.

The NPV of different replacement decisions/options is calculated and the one that offers the highest

returns. Positive AEV or the lowest negative AEV is selected.

Illustration

Dibco Ltd. is a manufacturing company which makes a wide range of products. One of these products

requires the use of a special machine.

The current policy of Dibco Ltd. is to replace each machine at the end of its useful life of four years.

The directors of the company arc considering whether to replace the machine more frequently due to

the fact that its productivity declines and running costs increase as it gets older.

There is insufficient demand for the company's products manufactured using the machine to justify

purchase of a second machine.

Dibco Ltd. sells the products made by the machine at Sh.12.00 each at which price it is able to sell up

to 500,000 units per annum.

Variable costs, excluding machine depreciation and running costs, amount to Sh.4.00 per unit.

Details of productive capacities and running costs of the machine are as follows:

Year of machine life Productive capacity

(units)

Running cost

Sh. “000”

1

2

3

4

500,000

500,000

400,000

400,000

600

650

750

900

The cost of buying the machine is Sh.6, 000, 000. The resale values of the machine are Sh.4, 000,000

for a one-year old machine, Sh.2, 500,000 for a two-year old machine, Sh. 1,000,000 for a three-year

old machine and zero for a four-year old machine. The company provides depreciation for its non-

current assets using the straight line method.

All costs and revenues are paid or received in cash at the end of the year to which they relate except

the initial cost of the machine which is paid immediately on purchase. The company has an annual

cost of capital of 10%.

Page 54

Required:

Advise the directors of Dibco Ltd. on whether to replace the machine every one, two, three or four

years.

Solution

Replacement after one year

NPV = PV of cash inflows – Initial Outlay

Cash inflows 500,000 x (12-4) – 600,000 + 4,000,000 = Sh. 7,400,000

NPV = 7,400,000 x PVIF1yr10% - 6,000,000

6,727,340 – 6,000,000

Sh. 727,340

= 727340

𝑃𝑉𝐼𝐹𝐴10%1𝑦𝑟 =

727340

0.9091

= 800,065.9993

Replacement after every two years

End of year Cash flows

Sh.

Discount factor

10%

Present value

Sh.

0

1

2

2

(6,000,000)

3,400,000

3,350,000

2,500,000

1.000

0.9091

0.8264

0.8264

Net present value

(6,000,000)

3,090,940

2,768,440

2066000

1925380

Standard NPV = 𝑁𝑃𝑉

𝑃𝐼𝑉𝐼𝐹𝐴10%𝑛𝑦𝑟𝑠

= 1925380

1.7355 = 1,109,409.39

Replacement after every three years

End of year Cash flows

Sh.

Discount factor

10%

Present value

Sh.

0

1

2

3

3

(6,000,000)

3,400,000

3,350,000

2,450,000

1,000,000

1.000

0.9091

0.8264

0.7513

0.7513

Net present value

(6,000,000)

3090940

2768440

1840685

751300

_______

2,452,365

Standard NPV = 𝑁𝑃𝑉

𝑃𝐼𝑉𝐼𝐹𝐴10%𝑛𝑦𝑟𝑠

Page 55

= 2452365

2.4869 = 986113.233

Replacement after every four years

End of year Cash flows

Sh.

Discount factor

10%

Present value

Sh.

0

1

2

3

4

(6,000,000)

3,400,000

3,350,000

2,450,000

2,300,000

1.000

0.9091

0.8264

0.7513

0.6830

Net present value

(6,000,000)

3090940

2768440

1840685

1,570,000

3,270,065

Standard NPV = 𝑁𝑃𝑉

𝑃𝐼𝑉𝐼𝐹𝐴10%𝑛𝑦𝑟𝑠

= 3270065

3.1699 = 1,031,898.78

A replacement should be made after every two years.

Illustration

ABC Ltd. is contemplating a replacement cycle for new machinery. This new machinery will cost Sh.

100 million purchase. The operating and maintenance costs for the future years are as follows:

Year 0 1 2 3

Operating and

maintenance costs (Sh.

“000”) 0 120,000 130,000 140,000

The resale values of the machinery in the second hand market are as follows:

Year 0 1 2 3

Resale value(Sh. “000”) 0 80,000 65,000 35,000

Assume:

1. The replacement is by an identical machine

2. There is no inflation, tax or risk

3. The cost of capital is 11%

Required;-

Advise ABC Ltd. on whether to replace this new machine on a one, two or three – year cycle.

Solution

Page 56

Replacement after 1 year

End of year Cash flows

Sh.

Discount factor

11%

Present value

Sh.

0

1

1

(100,000)

(120,000)

80,000

1.000

0.9009

0.9009

Net present value

(100,000)

(108,108)

_72,072_

(152,180)

Replacement after 2 years

End of year Cash flows

Sh.

Discount factor

11%

Present value

Sh.

0

1

2

2

(100,000)

(120,000)

(130,000)

65,000

1.000

0.9009

0.8116

0.8116

Net present value

(100,000)

(108,108)

(105,508)

(52,754)

(260,862)

Replacement after every 3 years

End of year Cash flows

Sh.

Discount factor

11%

Present value

Sh.

0

1

2

3

3

(100,000)

(120,000)

(130,000)

(140,000)

35,000

1.000

0.9009

0.8116

0.7312

0.7312

Net present value

(100,000)

(108,108)

(105,508)

(102,368)

(25,592)_

(390,392)

Equivalent Annuity = 𝑁𝑃𝑉

𝑃𝑉𝐼𝐹𝐴𝑟% 𝑛𝑦𝑒𝑎𝑟𝑠

Year 4

Replacement after 1 year = (136,036

𝑃𝑉𝐼𝐹𝐴 11% 1𝑦𝑟 =

(136,036)

0.9009 = Sh. (15100m)

2 years = (260,862)

𝑃𝑉𝐼𝐹𝐴 11% 2𝑦𝑟𝑠 =

(260,862)

1.7125= Sh. (152,328.1752)

3 year = (390,392)

𝑃𝑉𝐼𝐹𝐴 11% 3𝑦𝑟𝑠 =

(390,392)

2.4437 = Sh. (159754.4707)

ABC Ltd should replace the machine in a one-year cycle since it offers less cost.

Illustration

Page 57

Cosmos is evaluating two investments as follows: -

This is an investment in new machinery to produce a recently developed product. The cost of the

machinery which is payable immediately is sh.1.5 million, and the scrap value of the machinery at the

end of four years is expected to be sh.100,000. Capital allowances (tax-allowable depreciation) can be

claimed on this investment on a 25% reducing balance basis. Information on future returns from the

investment has been forecast to be as follows: -

Year 1 2 3 4

Sales volume(units/year)

Selling price (sh. unit)

Variable cost (sh. unit)

Fixed costs (sh./year)

50,000

20.00

10.00

105,000

95,000

24.00

11.00

115,000

140,000

23.00

12.00

125,000

75,000

23.00

12.50

125,000

This information must be adjusted to allow for selling price inflation of 4% per year and variable cost

inflation of 2.5% per year. Fixed costs, which are wholly attributable to the project, have already been

adjusted for inflation. Ridag Co pays profit tax of 30% per year one year in arrears.

Project 2

Cosmos plans to replace an existing machine and must choose between two machines. Machine 1 has

an initial cost of sh.200, 000 and will have a scrap value of sh. 25,000 after four years. Machine 2 has

an initial cost of sh.225, 000 and will have a scrap value of sh.50, 000 after three years. Annual

maintenance costs of the two machines are as follows:

Year 1 2 3 4

Machine 1 (sh./year)

Machine 2 (sh./year)

25,000

15,000

29,000

20,000

32,000

25,000

35,000

Where relevant, all information relating to Project 2 has already been adjusted to include expected

future inflation.

Taxation and capital allowances must be ignored in relation to Machine 1 and Machine 2.

Other information

Cosmos has a nominal before tax weighted average cost of 12% and a nominal after-tax weighted

average cost of capital of 7%.

Required;-

(i)Calculate the net present value of Project 1 and comment on whether this project is financially

acceptable to Cosmos Company.

(ii)Calculate the equivalent annual costs of Machine 1 and Machine 2 and discuss which machine

should be purchased.

(iii)Critically discuss the use of sensitivity analysis and probability as ways of including risk in the

investment appraisal process, referring in your answer to the relative effectiveness of each method.

Page 58

Solution

Cosmos Company

Initial Outlay

Cost of machinery sh.1, 500,000

Annual after Tax cash flows

Year 1

Sh.

2

Sh.

3

Sh.

4

Sh.

5

Sh.

Selling price

Variable cost

Contribution margin

Sales volume

Total contribution

Less fixed costs

Before tax cash flow

Less Tax

After Tax Cash flow

Add dep tax shield:

Net after Tax Cash flows

Terminal value

Net cash flows

Discount factor @ 7%

Pv of cash inflows

Less initial outlay

NPV

20.8

(10.25)

10.55

50,000

527,500

(105,000)

422,500

_____

422,500

_____

422,500

_____

422,500

0.9346

24.96

(11.28)

13.68

95000

1,299,600

(115,000)

1,184,600

(126750)

1,057,850

112500

1,170,350

______

1,170,350

0.8734

2,705,398.3

(1,500,000)

1,205,398.3

23.92

(12.3)

11.62

140,000

1,626,800

(125000)

1501,800

(355380)

1,146,420

84375

1,230,795

_____

1,230,795

0.8163

23.92

(12.81)

11.11

75000

833250

(125000)

708,250

(450540)

257710

63281

320991

100,000

420991

0.7629

(212475)

(212475)

159844

(52631)

_____

(52631)

0.7130

Depreciation

Year 1

Sh.

2

Sh.

3

Sh.

4

Sh.

Beginning balance

Less: dep @ 25%

End balance

Tax shield @ 30%

1,500,000

(375,000)

1,125,000

337,500

1,125,000

(281,250)

843750

84375

843750

(210937.5)

632812.5

6328.1

632812.5

(532812.5)

100,000

159844

This project if financially acceptable as it has a positive NPV.

Equivalent annual costs =𝑁𝑃𝑉

𝑃𝑉𝐼𝐹𝐴𝑛𝑦𝑟𝑠𝑟%

Machine 1

Year 0

Sh.

1

Sh.

2

Sh.

3

Sh.

4

Sh.

Page 59

Initial cost

Maintenance costs

Net cash flows

Discount factor @ 12%

(200,000)

_____

(200,000)

1.0000

(25,000)

(25,000)

0.8929

(20,000)

(20,000)

0.7972

(32,000)

(32000)

0.7118

25,000

(35,000)

(10,000

0.6355

NPV = (267,399.1)

EAC = (267,399.1)

𝑃𝑉𝐼𝐹𝐴4𝑦𝑟𝑠12% = sh. (88,036.991)

Machine 2

Year 0

Sh.

1

Sh.

2

Sh.

3

Sh.

Initial cost

Maintenance costs

Net cash flows

Discount factor @ 10%

(225,000)

_____

(225,000)

10000

(15000)

(15000)

0.8929

(20,000)

(29000)

0.7918

50,000

(25,000)

25,000

0.7118

NPV = sh. (236542.5)

EAC = (236542.5)

𝑃𝑉𝐼𝐹𝐴3𝑦𝑟𝑠 12% = Sh. (98485.51087)

Machine 1 has the lowest equivalent annual cost, thus should be purchased.

1.6 REAL OPTIONS

These are also known as managerial options or capital budgeting options. They are available to

finance managers when making capital budgeting decision.

Exam focus area

A real option relates to project appraisal. In previous questions, we have assumed that the only choice

available to us is to accept or reject the project based on the expected cash flows.

However, as will be explained below, it may be possible to improve the potential return by having the

right to change something about the project during its life. This would be a ‘real’ option. In the exam,

you are expected to be aware of the different types of ‘real’ options that might exist, and to be able to

value them using the Black Scholes model.

Types of real options

In order to explain the different types of real options, we will list them in turn together with a brief

illustration of the idea.

Option to delay

Page 60

Suppose we are considering a project, but the returns are uncertain because of forecast general

economic problems over the next few years.

The ability to delay starting the project could be attractive because if economic conditions turn out to

be unfavourable we could cancel, whereas if they turn out to be favourable we could go ahead and

maybe get even better returns.

The fact that we would be able to remove the ‘downside’ potential would mean that we had an option

and this would be worth paying for.

It would effectively be a call option (the right to invest in the project at a future date) and we could

use a formula to value it.

Option to expand

This would be similar to an option to delay in that we could invest a certain amount in the project now

and decide later whether or not to invest more (when we find out how successful the project is).

Again, this right would be worth money to us and could be valued, as a call option.

Option to abandon

When appraising (for example) a 5 year project, we usually assume that the project lasts for the full 5

years. However, if the cash flows turned out to be lower than expected, we would clearly want to be

able to consider stopping the project early.

Yet again, this right would effectively be an option – although this time a put option.

Illustration 1

Consider a project with the following characteristics

Year Cash flow sh. ‘000’ Abandonment value sh.

‘000’

0

1

2

3

4

(18000)

9000

8000

6000

7000

-

6000

5000

9000

-

The cost of capital of the company is 10%.

Required;-

Advice the management whether or not abandonment is a viable option and when to abandon.

Solution

Page 61

Step one – Ignoring the option to abandon, compute the NPV

Year Cash flow sh. ‘000’ PVIAF 10% Discounted Cash flow

0

1

2

3

4

(18,000)

9,000

8,000

6,000

7,000

1.000

0.9091

0.8264

0.7513

0.6830

(18,000)

8,181.9

6,611.2

4,507.8

4,781

6,081.9

Step II – NPV option

Option 1: If abandonment was at the end of year 1.

9000 x 0.9091 + 6000 x 0.9091 – 18000 = -4363.5

Option 2: If abandonment was at the end of year 2

NPV = 9000 x 0.9091 + 8,000 x 0.8254 + 5,000 x 0.8264 – 18,000

= 925.1

Option 3: If abandonment was at the end of year 3.

NPV = 9,000 x 0.9091 + 8,000 x 0.8264 + 6,000 x 0.7513 + 9,000 x 0.7513 – 18,000

= 8,062.5

Advice: Abandonment is a viable option because the NPV arising if abandonment was to be done of

sh.8062.8 is more than NPV if abandonment is to be ignored of 6081.9.

The optimal abandonment period is at the end of year 3 because this is the point where NPV is

maximized.

Option to redeploy

A firm may have decided to invest a considerable amount in equipment, staff, training etc. to

commence teaching CPA courses, on the basis that currently they appear to be the most profitable use

of the resources. However, projections could turn out to be wrong and it could be beneficial to

effectively stop the project earlier than planned and use the resources to teach some other

qualification.

This ability would be a put option (and the option to abandon is a special case of this).

Example 1

Warsaw company is considering a new project which requires an outlay of sh.10 million and has an

expected net present value of sh.2 million.

However, the economic climate over the next few years is thought to be very risky and the volatility

attaching to the net present value of the project is 20%.

Page 62

Warsaw is able to delay commencing the project for three years.

The risk free rate of interest is 6% p.a.

Estimate the value of the option to delay the start of the project for three years, using the

Black Scholes option pricing model.

Solution

P = current P.V. of project = sh.12 M [NPV = PV – Initial outlay therefore PV = NPV + Initial outlay)

E = capital expenditure = sh.10M

t = 3 years

r = 6%

σ = 20%

d1=𝐼𝑛(𝑃/𝐸)+(𝑟+0.5𝜎2)𝑇

𝜎√𝑇

d1 = 𝐼𝑛(

12

10)+ (0.06+0.5 𝑥(0.2)2)𝑥 3

0.20 𝑥√3

= 0.1823+0.24

0.3464 = 1.22

d2 = 1.22 – 0.2 x √3 = 0.87

N(d1) = 0.5 + 0.3888 = 0.8888

N(d2) = 0.5 + 0.3078 = 0.8078

C = Pnd1- 𝐸𝑛𝑑2

𝑒𝑟𝑡

C= 12 x 0.8888 – 10 × 0.8078

𝑒0.06×3

= sh.3.92M

Therefore the total project value = NPV of project without option value + Option value

= 2 + 3.92 = sh.5.92m

Strategic Investment option – This is where the management undertakes a project irrespective of the

net present value since undertaking the project can give rise to new opportunities.

REPLACEMENT ANALYSIS

Decision regarding replacement of an existing asset with another is based on the net present value and

internal rate of return of the incremental cash flows, i.e. the difference between periodic net cash

flows if the existing asset is kept and the periodic net cash flows if the asset is replaced.

In capital budgeting and engineering economics, the existing asset is called the defender and the asset,

which is proposed to replace the defender, is called the challenger. Estimation of incremental cash

C = Value of a call option

Page 63

flows for such replacement analysis involves calculation of net cash flows of the defender, net cash

flows of the challenger and then finding the difference in cash flows for both the assets.

Technique

Calculating periodic cash flows of existing asset is straightforward. Since the existing asset is already

purchased, the initial investment outlay is zero and the periodic net cash flows are calculated based on

the following formula:

Net cash flows = (revenue – operating expenses – depreciation) * (1 – tax rate) + depreciation.

If the asset is replaced, it involves investment is the new asset and sale or disposal of the existing

asset. Disposal of exiting asset has some income tax implications, which need to be reflected in the

calculation of initial investment as follows:

Initial investment after replacement = cost of new asset - sale proceeds of old asset +/- tax on disposal.

Tax on disposed asset = (sale proceeds of old assets – book value of old asset) * tax rate

As evident from the equation above, if the old asset is sold at an amount higher than its book value,

the company bears a related tax cost, which is added to the initial investment. Similarly, if the sale

proceeds are lower than the book value of the asset sold, there is a resulting tax shield, which is

subtracted from sum of cost of new asset and sale proceeds of the old asset.

Components of cash flows on replacement decisions.

Incremental initial investment cost (end of year 0)

Sh. ‘000’

Cost of new project

Incidental costs: Institution

Insurance on transit

Transport to site

The total of new project

Less: Current market value of old project (disposal value of old project)

Add: Investment in working capital (CA – Cl)

Incremental initial investment

xx

xx

xx

xx

xxx

xx

xxx

xx

xxx

Incremental Annual after tax cash flows (End of year 1)

Year 1 2 3 4 5

Revenue

Less variable costs

Contribution

Xxx

(xx)

xxx

xxx

(xx)

xxx

xxx

(xx)

xxx

xxx

(xx)

xxx

xxx

(xx)

xxx

Page 64

Less fixed cost

Before tax cash flows

Less tax

After tax cash flows

Add depreciation tax shield of

new machine

Less depreciation tax shield of

old machine

Net after tax cash flows

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

Xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

(xx)

xxx

Incremental terminal cash flows

Sh.

Salvage value of the new machine

Less salvage value of the old machine

Add release/recovery of working capital

xxx

(xxx)

xxx

xxx

xxx

Illustration

Kisasi Company is considering buying a new machine in order to produce a new product.

The machine will cost sh.1,800 and is expected to last for 5 years at which time it will have an

estimated scrap value of sh.1,000.

They expect to produce 100,000 units p.a. of the new product, which will be sold for sh.20 per unit in

the first year.

Production costs per unit (at current prices) are as follows:

Materials = sh.8

Labour = sh.7

Materials are expected to inflate at 8% p.a. and labour is expected to inflate at 5% p.a.

Fixed overheads of the company currently amount to sh.1,000. The management accountant has

decided that 20% of these should be absorbed into the new product.

The company expects to be able to increase the selling price of the product by 7% p.a.

An additional sh.200 of working capital will be required at the start of the project.

Capital allowances: straight line method.

Tax: 25%, payable immediately

Cost of capital: 10%

Required:

Calculate the NPV of the project and advise whether it should be accepted.

Solution

Cost = 1,800

Page 65

N = 5 years

Scrap value = 1,000

Production = 100 p.a

To be sold @ 20/unit for 1st year

Depreciation = 1,800−1,000

5 = 160

Depreciation tax shield = 160 x 25% = 40

Initial outlay (Io)

Cost of machine = 1,800

Investment in working capital 200

Io 2,000

CASH FLOW STATEMENT (SH.000)

Production Y1 Y2 Y3 Y4 Y5

Sales

Less material cost

Labour

Fixed cost (20% x 1000)

EBTD

Less tax 25%

Add Depreciation tax shield

(1800−1000

5)× 0.25

PIVIF10% yrs

pv

2000

(864)

(735)

(200)

201

(50.25)

150.75

40

190.75

0.9091

173.41

2,140

(933.12)

(771.75)

(200)

235.13

(58.7825)

176.3475

40

216.3475

0.8264

178.79

2,289.8

(1007.7696)

(810.3375)

(200)

271.6929

(67.923225)

203.769675

40

243.769675

0.7513

183.14

2,450.086

(1088.391168)

(850.854375)

(200)

310.840457

(77.71011425)

233.1303428

40

273.1303428

0.6830

186.55

2,621.59202

(1175.462461)

(893.3970938)

(200)

252.7324662

(63.18311655)

189.5492497

40

229.5493497

0.6808

156.28

Terminal cash flows

Scrap value of new – 1000

Add released of working capital – 200

1,200×0.6808 = 816.96

NPV of the project

Total PV = PV of operational cashflows + PV of terminal cashflows.

= 878.17 + 816.96 = 1695.13

NPV = 1695.13 – 2000 = -304.87

Advise: Accept the project because it has a positive NPV

Illustration II

Chuma Ltd is considering replacing a machine. The existing machine was bought 3 years ago at a

price of sh.50 million. The machine is expected to have a useful life of 5 more years with no scrap

value at the end of its useful life. The machine could be disposed of immediately at sh.35 million. The

new machine will cost sh.80 million with a useful life of 5 years and an expected terminal value of

Page 66

sh.5 million. With the introduction of the new machine; sales are expected to increase by sh.25

million per annum over the next five years.

The contribution margin is expected to be 40% and the corporate tax rate is 30%. The operation of the

new machine will also require an immediate investment of sh.8 million in working capital. Installation

costs of the new machine will amount to sh.6 million.

Depreciation is to be provided for on a straight line basis. The company’s cost of capital is 12%.

Required:

Advise the management of Chuma Ltd on whether to replace the machine.

Solution

Chuma Limited

Incremental Annual after Tax Cash flows

1 – 5 years’ depreciation is on straight line

Sh. ‘m’

Annual increase in sales

Annual contribution increase @ 40%

Less increase in tax cash flow @ 30%

Add depreciation tax shield increase

New machine = 16.2 (86 – 5) ÷ 5

Old machine = (6.25)

9.95 ×30%

Annual after tax cash flow income

25

10

(3)

7

2.985

9.985

Incremental Terminal Cash flow (end of years)

Sh. ‘m’

Scrap value of new machine 5

Sh. ‘m’

Incremental initial investment cost

Cost of new machine

Add installation costs

Total cost of new machine

Less disposal value of old machine

Tax gain/(loss) on disposal

Disposal value of old machine 35

NBV of old machine

50,000-(50,000-0) * 3 (31.25)

6 3.75

Tax on gain 30%* 3.75

Add investment cost in working capital

Initial investment cost

80

6

86

(35)

1.125

8

60.125

Page 67

Less scrap value of old machine

Add release of working capital

(0)

8

13

Net Present Value = PV of Cash inflows – Initial Outlays

= 9.985×PVIFA 5yrs12% + 13 ×PVIF 5 yrs 12% - 60.125

9.985×3.6048 + 13 ×0.5674 – 60.125

35.993928 + 7.3762 – 60.125 = (Sh. 16.754872m)

Chuma Ltd should not replace the machine, as it would result into a negative NPV thus minimising

the shareholders’ wealth.

Page 68

CHAPTER TWO

PORTFOLIO THEORY AND ANALYSIS CHAPTER KEY OBJECTIVES

To be able to understand the following

1. The modern portfolio theory: background of the theory; portfolio expected return; the actual and

weighted portfolio risk; derivation of efficient sets; the capital market line (CML) model and its

applications, the mean variance dominance rule; short comings of portfolio theory

2. Capital Asset Pricing Model-CAPM : background of the theory; assumptions; beta estimation -

beta coefficient of an individual asset and that of a portfolio and the interpretation of the result;

security market line(SML) model and its applications; conceptual differences between portfolio

theory and capital asset pricing model

3. Shortcomings of the capital asset pricing model

4. The Arbitrage pricing model (APM) and other multifactor models: background of the theory;

conceptual differences between the Capital asset pricing model and the Arbitrage pricing model;

application of the Arbitrage pricing model, shortcomings of Arbitrage pricing model; Pastor

Stambaugh model

5. Evaluation of portfolio performance: Treynor’s measure, Sharpe’s measure, Jensen’s measure,

appraisal ratio measure, information ratio, Modigliani and Modigliani (M2)

2.1 THE MODERN PORTFOLIO THEORY

A portfolio refers to a combination of investments held together by an investor in a firm.

Portfolio may include:

- Financial assets /securities e.g. ordinary shares, preference shares, debentures or bonds.

- Physical/tangible assets i.e. equipment, real estate etc.

An investor shall combine the assets with the main objectives of minimising the overall risk and

maximising total returns.

A company will always experience two types of risk i.e.

- Financial risk

- Business risk

Financial Risk

It is the risk associated with the use of debt/capital with fixed returns in the company’s capital

structure.

It is also known as gearing/leverage risk and is measured by use of gearing ratio i.e.

Gearing Ratio = 𝐶𝑊𝐹𝑅

𝐶𝑊𝐹𝑅+𝐶𝑊𝑉𝑅× 100

Page 69

CWFR – Capital with fixed returns (long term debt /debentures

CWVR – Capital with variable returns (ordinary shares i.e. common equity capital)

Business Risk/Operating Risk

This is a combination of;

a) Unsystematic Risk

b) Systematic Risk

Unsystematic Risk

It is a risk that is unique to individual firms in the industry and can be eliminated/reduced through

diversification of investment portfolio.

Due to its diversifiable nature, it is therefore not incorporated in investment appraisal since a company

can avoid it e.g. poor employee remuneration, industrial strikes, poor marketing.

Unsystematic risk factors don't affect everyone; indeed, their impact may be unique to an individual

company or restricted to a small number of companies, with some being winners and some being

losers. For example, the weather – if we have a wet summer then raincoat manufacturers will benefit

but sunglasses manufacturers will suffer. However, for the majority of businesses, it will not make

any difference. Overall, the stock market is unlikely to be affected much by the weather.

Can be measured by the standard deviation

Systematic Risk

It refers to variability in returns of securities cash flows due to factors that affect the firms in the

industry e.g. political instability, inflation, higher lending rates etc.

Due to its undiversifiable nature, it is therefore incorporated in investment appraisal and it is

measured using beta factor 𝛽.

Systematic risk will affect all companies in the same way (although to varying degrees). For example,

the vast majority of companies suffer in a recession but not necessarily to the same extent – e.g.

house-builders typically suffer more than bakers do.

This explains why diversification works, typically there will be winners and losers regarding a

particular risk factor but when combined in a portfolio, the impact is cancelled out. Diversification

can almost eliminate unsystematic risk, but since all investments are affected in the same way by

macroeconomic i.e. systematic factors, the systematic risk of the portfolio remains.

The ability of investors to diversify away unsystematic risk by holding portfolios consisting of a

number of different shares is the cornerstone of portfolio theory.

Page 70

Illustration

Systematic /Unsystematic Risk

Importance of holding a portfolio rather than a single security

It helps in risk diversification

Risk diversification is only possible if the returns of two investments are negatively correlated.

Correlation is a statistical measure of how strong two variables are related.

The greater the negative correlation, the greater the degree of risk reduction.

To maximise the overall returns on an investment

Through diversification, investors attempt to achieve the highest returns at the lowest risk level.

It leads to tax savings

Different securities generate different returns which are taxed separately/differently

A portfolio can therefore be used to reduce the investors WACC.

It leads to growth of investment.

The returns generated from a diversified/portfolio can be reinvested leading to an

increase/expansion of an investment.

Unsystematic

Risk

Systematic

Risk

No. of Securities

Risk

Business Total

Operating Risk

Page 71

It promotes marketability and liquidity of an investor’s assets.

Securities in the portfolio tend to have a ready market due to its publicity and can therefore be

easily converted into cash without losing value.

Weaknesses of Portfolio Theory

1. It considers total risk as measured by standard deviation. In real sense, some risk is diversified

once a portfolio is formed.

2. It uses historical data in analysing the portfolios.

More so, the associated probabilities are not objectively determined.

3. It ignores economies of scale that may arise if more securities are combined together.

It only incorporates the risk reduction and the maximisation of returns.

4. It uses indifference curves in measuring the satisfaction of investors (investors utility) which is

not always practical. It is too theoretical.

5. It assumes that all investors are risk averse such that they would only be willing to increase their

investments if they are promised some returns.

6. It assumes that all projects are divisible. But in reality some projects are indivisible.

Factors Affecting Efficiency of a Good Portfolio

i) Number of securities forming the portfolio

The large the number of securities, the more efficient the portfolio would be.

ii) Correlation between securities in the portfolio i.e. how related the securities in a portfolio are.

iii) The proportions/weight of each security in the portfolio.

In measuring correlation between returns of two securities, co-variance is used.

COV (A, B) = ∑ (RA – ERA) (RB – ERB) P

COV (A, B) = Co-variance between securities A & B

ERA Expected Returns of Security A

ERB Expected Returns of Security B

RA Returns of A

RB Returns of B

Covariance would be negative for a negative correlation (Good portfolio) and it would be positive for

a positive correlation (Inefficient portfolio).

To measure the strength of the association between returns of different securities, we use correlation

coefficient ℓ (A,B)

ℓ(A,B) = 𝐶𝑂𝑉 (𝐴,𝐵)

𝛿𝐴𝑥𝛿𝐵

NB

(i) If ℓ (A,B) is negative, risk would be reduced when a portfolio is formed.

(ii) If ℓ A,B is positive, portfolio formation would not reduce the risk.

Page 72

(iii) If no correlation then risk reduction is 100% efficient.

Portfolio risk refers to the possibility that the actual returns on an investment may vary from expected

returns.

Variance and standard deviation are the most common measures of portfolio risk.

The risk of a portfolio depends on

a) Covariance between returns between securities

b) Risk of individual security that forms the portfolio as measured by the standard deviation

Portfolio return is a weighted average of the returns of given securities that form the portfolio i.e.

Expected portfolio return = ERAWA + ERBWB + ERnWn

ERP = Expected Return of a Portfolio

ERA/ERB/ERn= Expected returns of securities A, B& n that form the portfolio.

WA,WB,Wn – Weight/Proportion of A, B & n in the portfolio

Portfolio Risk = 𝛿P = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒

Variance of portfolio = ∑𝛿𝐴2𝑊𝐴

2 + 𝛿𝐵2𝑊𝐵

2 + 2𝑊𝐴𝑊𝐵𝐶𝑜𝐴, 𝐵

𝛿P = √∑𝛿𝐴2𝑊𝐴

2 + 𝛿𝐵2𝑊𝐵

2 + 2𝑊𝐴𝑊𝐴𝐶𝑜𝐴, 𝐵

Where: 𝛿P = Portfolio Risk (Standard deviation of portfolio)

𝛿𝐴2&𝛿𝐵2 = Variance of securities A & B.

WA& WB = Weight of Securities A & B

COV A, B = Covariance between returns of securities A & B

Efficient /Optimum Portfolios

Investors have different risk attitudes

Risk takers would prefer higher risk to obtain higher returns while risk averse investors would prefer

lower risks, which is associated with lower returns.

In the general market a portfolio can offer highest return at a given level of risk while another

portfolio can offer the lowest risk at a given level of return.

Portfolio risk is measured by the standard deviation.

Illustration

A

B

C

D

J

I

G

H

F E

E.F.C

Porfolio

Return

Page 73

Portfolios A, E, F & G promise the same level of return at different risk levels, however portfolio A

has the lowest risk hence it dominates the rest.

Consequently, portfolios B & C dominates H & I, J respectively.

Portfolios D, J, I & G promises the same level of risks at different levels of returns. However,

portfolio D promises the highest returns hence it dominates portfolios H, I & G.

In addition, portfolio C dominates H & F and portfolio B dominates it.

The portfolios that dominates others in the market in terms of risk and returns form an efficient set

and are joined together by a conceive curve known as Efficient Frontier Curve E.F.C.

All portfolios that are below EFC are said to be inferior

portfolios since they don’t offer an

optimum risk return trade off.

Utility of investors is obtained from

consumption of returns and is

normally determined using

indifference curves.

The higher the indifference the

greater the utility derived.

Indifference Curves

Page 74

The most optimum portfolio falls at a point where the indifference curve is a tangent with efficient

frontier curve.

EFC – Efficient Frontier curve

Page 75

EFC is a tangent at portfolio M. This portfolio is known as market portfolio.

It offers the highest return per unit of risk.

Portfolios A, B, C, D& E are therefore efficient portfolios while portfolio M is the market optimum

portfolio.

Capital Market Line CML

In absence of risk free securities, all efficient portfolios would be found along efficient frontier curve.

However, some securities without any form of risk (risk free securities i.e. Treasury bills and bonds

may form part of the portfolio.

If an investor has a combination of risky and risk free assets, a line extending from risk rate of return

to a point of tangent on the efficient frontier curve is known as a capital market line.

This line indicates the risk and return relationship of a portfolio comprising both risky and riskless

securities.

Portfolio Risk ( )

Portfolio

Return C.M.L

EFC

RF

RM

Page 76

The difference between expected market return and risk free rate of return is the extra return an

investor receives for accepting additional risks (Risk Premium)

The capital market line equation is an equation in the form of y = a + bx

Where

Y = dependent variable required rate of return

A – y when x = 0, risk free rate

B – gradient Δ𝑦

Δ𝑥,

The Risk Premium/Unit of Risk can be determined by calculating the gradient of capital market line

i.e.

Gradient = ∆𝑦

∆𝑥 ,

𝑅𝑀−𝑅𝐹

𝛿𝑚

Risk Premium = 𝑅𝑀−𝑅𝐹

𝛿𝑚

% of Risk Premium = [𝑅𝑀−𝑅𝐹

𝛿𝑀] 𝛿𝑃

Required Returns = RF + Risk Premium

Required Portfolio Return = RF + [𝑅𝑀−𝑅𝐹

𝛿𝑀] 𝛿𝑃

RF – Risk Free Rate of Return

RM – Required Market Return

𝛿𝑀 – Standard deviation of the market

𝛿𝑃 – Standard deviation of the portfolio

The above equation is known as Capital Market Line Equation.

Conclusion

If a portfolio falls above the capital market line such portfolio is undervalued.

If a portfolio falls below the capital market line, such portfolio is overvalued.

If a portfolio falls along the capital market line, such portfolio is correctly valued.

An undervalued portfolio is a super-efficient portfolio overvalued portfolio is an inefficient portfolio

while correctly portfolio is efficient portfolio.

Illustration

An investor has the choice of the following share investments:

Share

Risk (σ) Expected return

Page 77

A 20% 8%

B 25% 6%

C 23% 4%

D 20% 2%

E 22% 2%

Which share (or shares) will the investor definitely not choose?

The investor cannot choose portfolio A since it has a lower return and higher risk.

Illustration 2

Janis currently has a portfolio of shares giving a return of 18% with a risk of 10%. He is considering

investing in one of the following additional investments.

A B

Return 8% 8%

Risk 5% 3%

Coefficient of correlation with existing portfolio -0.7 +0.4

The new investment will comprise 20% of his enlarged portfolio.

Which of the two investments should he choose?

Choose investment A since the returns are negatively correlated with existing portfolio hence

diversifying risk.

Illustration 3

6 portfolios are available for investment with the following characteristics.

Portfolio ERP% 𝜹𝑷

A

B

C

D

E

F

25

16

19

32

89

22.5

12

6

8

16

2

10

Expected Return on the market portfolio is 12% with a standard deviation of 4%.

The risk free rate of return is 5%.

Required;

(i)Using CML, advice the investor on the portfolio that is undervalued, correctly valued and

overvalued.

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(ii)In case of an inefficient portfolio in (i) above, determine the standard deviation that should be

achieved for efficiency to be arrived at.

Solution

CML = RF + [𝑅𝑀−𝑅𝐹

𝛿𝑀] 𝛿𝑃

Required Return ERP Comment

A = 5 + [12−5

4] 12 = 26%

B = 5 + [12−5

4] 6 = 15.5%

C = 5 + [12−5

4] 8 = 19%

D = 5 + [12−5

4] 16 = 33%

E = 5 + [12−5

4] 2 = 8.5%

F = 5 + [12−5

4] 10 =22.5%

25%

16%

19%

32%

8.9%

22.5%

Overvalued

Undervalued

Correctly valued

Overvalued

Undervalued

Correctly valued

Portfolios A and D are inefficient since they are overvalued. The standard deviation of these

portfolios that would make them efficient is:

A 5 + [12−5

4] x = 25

5 + 175x = 25 𝛿𝑃 1.75𝑥

1.75 =

25−5

1.75 x = 11.43%

D 5 + [12−5

4] x = 32 𝛿𝑃

1.75𝑥

1.75 =

32 −5

1.75 = x = 15.43%

Mean-Variance Analysis

Mean-variance analysis is the process of weighing risk, expressed as variance, against expected

return. Investors use mean-variance analysis to make decisions about which financial instruments to

invest in, based on how much risk they are willing to take on in exchange for different levels of

reward. Mean-variance analysis allows investors to find the biggest reward at a given level of risk or

the least risk at a given level of return.

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Return per unit of risk = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑟𝑒𝑡𝑢𝑟𝑛

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

Minimum variance portfolio

Definition: A minimum variance portfolio indicates a well-diversified portfolio that consists of

individually risky assets, which are hedged when traded together, resulting in the lowest possible risk

for the rate of expected return.

If we want to find the exact minimum variance portfolio allocation for these two assets, we can use

the following equation:

Wx = 𝜹𝟐𝒚−𝑪𝒐𝒗𝒙𝒚

𝜹𝟐𝒙+𝜹𝟐𝒚−𝟐𝑪𝒐𝒗𝒙𝒚

Illustration

a) Mr Akili Mingi holds the following portfolio of four risky assets and a deposit in a risk free asset.

The table below shows the respective portfolio weightings and the current returns on the assets,

together with their beta coefficients.

Asset Weighting (%) Current

returns (%)

Beta coefficient

A

B

C

D

Risk - free asset

20

10

15

25

30

12.0

18.0

14.0

8.0

5.0

1.5

2.0

1.2

0.9

0.0

The overall return on the market portfolio of risky assets is 11 % and this is expected to continue for

the foreseeable future.

Required:

(i)The portfolio current return and the portfolio beta.

(ii)Determine the assets which are inefficient, efficient or super-efficient.

(iii)In view of your answer in (a)(ii) above, predict how the future asset values and, hence, their rates

of return would behave as the market moves towards full equilibrium.

b) A fund is split between two securities X and Y. The following data relate to these securities:

Variance for asset Y = σy2= 297.6

Covariance (COVx, y) =54

Variance for asset X = σ2x= 10

Required:

The proportions that an extremely risk-averse individual would place in a portfolio comprising assets

X and Y to obtain a minimum standard deviation.

Page 80

Solution

(a)

(i) The portfolio current return

The portfolio return is a weighted average of the individual asset return.

i.e. ∑WR – W is the weight and R is the asset return

Rp= ∑current returns of individual securities × weights

Rp= 0.2 × 12 + 0.1 × 18 + 0.15 ×14 + 0.25×8 + 0.3 ×5 =9.8%

The portfolio Beta

The portfolio Beta is a weighted average of the individual assets.

Rp= ∑weights × beta of individual securities

Rp= 0.2 × 1.5 + 0.1 × 2 + 0.15 ×1.2 + 0.25×0.9 + 0.3 ×0 =0.905

0.905<1

Systematic risk of the portfolio is smaller than that of market portfolios i.e. returns of asset in this

portfolio are less sensitive to changes in returns of market portfolio. Implying that this is a portfolio

for risk averse individuals.

(ii) Assets which are inefficient, efficient and super-efficient

HINT;-

Compute Alpha values of each security and if ;-

𝛼 is negative – then the security is inefficient.

𝛼 is positive –then the security is super efficient.

𝛼 is 0– then the security is efficient.

Asset Market

Premium

(%)=(Rm-Rf)

11%-5%

Required

return %

(CAPM Returns)

Actual

return(R)

𝜶=

R- CAPMR

%

Assessment

A

B

C

D

6%

6%

6%

6%

5 +(1.5×6) = 14%

5 + (2.0×6) =17%

5 + (1.2×6) =12.2%

5 + (0.9×6) =10.4%

12%

18%

14%

8%

-2

1

1.8

-2.4

Inefficient

Super-efficient

Super-efficient

Inefficient

Super-efficient assets offer in excess of what their risk factors warrant. They have 𝛼 positive values.

iii) How the future asset values and hence their rates of return would behave as the market

moves towards full equilibrium

Super-efficient assets are very attractive because they offer abnormal returns. The demand for the

super-efficient assets will rise drastically implying that the owners would need a higher return (CAPM

returns). Since the expected return of the super-efficient assets will remain static in the short run, their

Page 81

profitability will reduce as their CAPM returns increase. This trend will continue until the Alpha

value of these super-efficient falls down to zero at equilibrium. The opposite of the above argument

will hold for in efficient assets.

(b) Proportions that an extremely risk averse individual would place in a portfolio comprising

assets X and Y to obtain a minimum standard deviation.

HINT

The lowest possible standard deviation is zero and therefore uses the following formulae for

computing optimal proportions.

WX = 𝜎2𝑦−𝐶𝑂𝑉𝑥,𝑦

𝜎2𝑥+𝜎2𝑦−2𝐶𝑂𝑉𝑥,𝑦

WY =297.6−(−54)

(10+297.6)−(2 ×−54)=

351.6

415.6= 0.846= 84.6%

And therefore

WY = 1- WX =10.846 = 0.154 = 15.4%

2.2 CAPITAL ASSET PRICING MODEL (CAPM)

CAPM focuses on systematic risk as measured by Beta coefficient whereas portfolio theory focuses

on total risk as measured by the standard deviation.

CAPM allows the analyst to split total risk of a security into two i.e.

Systematic Risk/Un-diversifiable Risk

Unsystematic Risk/Diversifiable Risk

It provides a framework for measuring systematic risk of an individual security by relating it with the

systematic risk of a well-diversified portfolio

Systematic risk is measured by (β) factor of a security.

Note:

The beta of a security is the sensitivity of security returns to changes in returns of the market

portfolio.

A security whose returns are highly correlated with fluctuations in the market is said to have a high

level of systematic risk. It does not have much risk-reducing potential on the investor’s portfolio and

therefore a high return is expected of it. On the other hand, a security which has a low correlation with

the market (low systematic risk) is valuable as a risk reducer and hence its required return will be

lower.

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The measure of the systematic risk of a security relative to that of the market portfolio is referred to as

its beta factor. In practice industries such as construction are far more volatile than others such as food

retailing in addition, would have correspondingly higher beta.

Note:

Beta of a security can be worked and compared with that of the market portfolio. If Beta coefficient is

more than one the security is known as Aggressive security implying that it has a higher systematic

risk as compared to market portfolio.

If Beta coefficient is less than one the security is known as defensive security implying that it has a

lower systematic risk compared to the market portfolio. If Beta is equal to one the security return will

follow general trend of stock market.

The CAPM shows the linear relationship between the risk premium of the security and the risk

premium of the market portfolio.

The same formula can be applied to compute the minimum required rate of return of a capital

investment project carried out by a company, because the company is just a vehicle for the

shareholders, who will view the project as an addition to the market portfolio.

In order to use the CAPM, investors need to have values for the variables contained in the model.

The beta of a security can be measured using the formula;

𝛽𝑗 =𝑐𝑜𝑣 (𝑅𝑗,𝑅𝑀)

𝛿2𝑀

B

Recall = COV (Rj, Rm) = ∑(Rj – ERj) (Rm – ERM)P

ℓj,m = 𝐶𝑜𝑣 (𝑅𝑗,𝑅𝑀)

𝛿𝑗 𝑥 𝛿𝑀

COV Rj, Rm = ℓj,m x 𝛿𝑗 𝑥 𝛿𝑀

𝛽j = ℓ𝑗,𝑚 𝑥 𝛿𝑗 𝑥 𝛿𝑚

𝛿2𝑀 =

ℓ𝑗,𝑚 𝑥 𝛿𝑗

𝛿2𝑀

Where: 𝛽𝑗 – Beta factor of security j

ℓ (j,m) – Correlation coefficient between returns of security j and the market.

𝛿𝑗, 𝑚 – Standard deviation of security j and the market.

The beta factor of the portfolio is the weighted beta of securities that form a portfolio i.e.

𝛽𝑃 =𝛽𝐴𝑊𝐴 + 𝛽𝐵𝑊𝐵 + ………… 𝛽𝑛𝑊𝑛

CAPM operates under the following assumptions

1. Capital markets are assumed to be perfect in that no transaction costs or taxes are incurred.

Page 83

2. Capital markets are assumed to be efficient in that security prices reflects all available

information.

3. Investors are assumed to be risk averse i.e. they would only take additional risk if they are assured

of adequate compensation.

4. Investment decisions are based on a single time period.

5. Expected returns of the portfolio are assumed to be normally distributed i.e. they take the form of

a straight line equation y = mx + c

RF = [Rm – Rf] 𝛽P

6. Investors are assumed to lend and borrow at risk free rate of return.

7. CAPM is a single factor model.

8. It is assumed that systematic risk is the only relevant risk since unsystematic risks has been

eliminated through portfolio building (diversification)

9. Investors expectations are homogeneous i.e. similar and identical in all aspects.

The implications of CAPM for project appraisal

1. If the shareholders of a company are well-diversified, then their shares in this

company are just part of their overall portfolio.

2. If the company is to invest the shareholders’ money in a new project, then the project

should be appraised in the same way as the shareholders themselves would appraise

the investment if they were invested their money in it directly.

3. If they were investing directly, then they would base their required return simply on

the β of that investment (not on how it related to any particular other investment in

their portfolio).

Therefore, when the company is appraising a new project they should calculate the β of

the project, determine the required return for that β, and appraise the project at that

required return.

Using CAPM in investment appraisal

The CAPM produces a required return based on the expected return of the market E(rm), the risk-free

interest rate (Rf) and the variability of project returns relative to the market returns (β). Its main

advantage when used for investment appraisal is that it produces a discount rate which is based on the

systematic risk of the individual investment. It can be used to compare projects of all different risk

classes and is therefore superior to an NPV approach which uses only one discount rate for all

projects, regardless of their risk.

The model was developed with respect to securities; by applying it to an investment within the firm,

the company is assuming that the shareholder wishes investments to be evaluated as if they were

securities in the capital market and thus assumes that all shareholders will hold diversified portfolios

and will not look to the company to achieve diversification for them.

Example

Required return

Page 84

Panda is all-equity financed. It wishes to invest in a project with an estimated beta of 1.5. The project

has significantly different business risk characteristics from Panda’s current operations. The project

requires an outlay of sh.10,000 and will generate expected returns of sh.12,000.

The market rate of return is 12% and the risk-free rate of return is 6%.

Required:

Estimate the minimum return that Panda will require from the project and assess whether the project

is worthwhile, based on the figures you are given.

Solution

We do not need to know Panda’s current weighted average cost of capital, as the new project has

different business characteristics from its current operations. Instead we use the capital asset pricing

model so that;

Required return = RF + (ERM – RF)Be

= 6 + (12 – 6)1.5 = 15%

Expected return = 12,000−10,000

10,000= 20%

Thus the project is worthwhile; as expected return exceeds required return i.e. the Alpha value is

positive (α)

α= ER – CAPMRs

= 20% - 15% = 5%

Limitations of using CAPM in investment decisions

The greatest practical problems with the use of the CAPM in investment decisions are as follows.

(a) It is hard to estimate returns on projects under different economic environments, market returns

under different economic environments and the probabilities of the various environments.

(b) The CAPM is really just a single period model. Few investment projects last for one year only and

to extend the use of the return estimated from the model to more than one time period would

require both project performance relative to the market and the economic environment to be

reasonably stable.

In theory, it should be possible to apply the CAPM for each time period, thus arriving at

successive discount rates, one for each year of the project’s life. In practice, this would exacerbate

the estimation problems mentioned above and also make the discounting process much more

cumbersome.

(c) It may be hard to determine the risk-free rate of return. Government securities are usually taken to

be risk-free, but the return on these securities varies according to their term to maturity.

(d) Some experts have argued that betas calculated using complicated statistical techniques often

overestimate high betas, and underestimate how betas, particularly for small companies.

Security Market Line (S.M.L)

Page 85

The trade-off between risk and return in a well-diversified portfolio is represented by a security

market line.

The SML is similar to capital market line except that:

CML deals with efficient portfolios while SML deals with individual securities in the portfolio.

CML deals with total risk as measured by the standard deviation while SML only deals with

systematic risk as measured by the beta factor.

SML can be represented graphically as follows:

SML equation is as follows:

Rj = RF + (RM – RF) 𝛽𝑗

Where Rj = security/Minimum required rate of return from security.

RF = Risk free Rate of Return

Rm = Expected market return (Return on the market)

𝛽𝑗 = Systematic risk of security/beta factor

Security Risk ( )

Security

Return S.M.L

M

RF

RM

M = Market Portfolio

= 1

Page 86

Illustration I

Betty Muye has invested 75% of her funds in shares of company X and 25% in shares of company Y.

The following probability distribution relates to the shares of the two companies:

State of economy Probability Return on company X

shares (%)

Return on Company Y

shares (%)

Boom

Steady growth

Stamp

0.2

0.6

0.2

24

12

0

5

30

-5

Required:

(i) Expected returns on the shares of companies X and Y

(ii) Standard deviation of returns on shares of companies X and Y

(iii) Coefficient of correlation between the returns on shares of companies X and Y.

(iv) Expected portfolio return.

(v) Portfolio risk

Solution

Expected Returns on the shares

X = 24 x 0.2 + 12 x 0.6 + 0 x 0.2 = 12%

Y = 5 x 0.2 + 30 x 0.6 + -5 x 0.2 = 18%

Standard deviation of returns on shares

𝛿 = √∑(𝑅 − 𝐸𝑅)2𝑃

X Y

(24 – 12)2× 0.2 =

(12 – 12)2× 0.6 =

(0 – 12)2× 0.2 =

28.8

0

28.8

57.6

(5 – 18)2× 0.2 (30– 18)2× 0.6

(-5 – 18)2× 0.2 =

33.8

86.4

105.8

226

8x = √57.6

= 7.59

8y = √226

= 15.03

Coefficient of correlation between the shares

ℓ(x,y) = 𝐶𝑜𝑣(𝑥,𝑦)

𝛿𝑗𝑥𝛿𝑀

Cov (x,y) = ∑(Rx – ERx) (Ry – ERy)𝑃𝑖

(24 – 12) (5 – 18) 0.2 =

(12 – 12) (30 – 18) 0.6 =

(0 – 12) (-5 - 18) 0.2 =

(31.2)

0

55.2

Page 87

24

ℓ(x,y) = 24

7.59 𝑥 15.03 = 0.21

ERP = ER x Wx + ERyWy

12 x 0.75 + 18 x 0.25 = 13.5%

𝛿P = √∑𝛿𝑥2𝑊𝑥2 + 𝛿𝑦2 𝑊𝑦2 + 2𝑊𝑥𝑊𝑦x 𝐶𝑜𝑋, 𝑌

√7.592 × 0.752 + 15.032 × 0.252 + 2 × 0.75 × 0.25 × 0.21

√32.405 + 14.119 + 0.079= 6.83

Illustration 2

a. The correlation coefficient between security S and that of the market is 0.96. the variance of

security is 4.36. determine the beta of security S if the variance of the market returns is 2.16.

b. If the risk free rate of return is 10% and the expected returns of the market is 14%, determine the

required rate of return of security S.

Solution

(a) 𝛽 = ℓ(𝑆,𝑀)𝑥 𝛿𝑆

𝛿𝑀

0.96 𝑥 √4.36

√2.16 = 1.3639

(b) RB = RF + (RM – RF)𝛽

10 + [14 – 10] 1.3639 = 15.4556%

Illustration 3

Galaxy Limited is an all equity financed company with a cost of capital of 18.5%. The

company is considering the following-capital investment projects:

Project Initial outlay

Sh. ‘000’

Expected cash flows in

one year.

Sh. ‘000’

Beta

A

B

C

D

1,000

1,000

1,500

2,000

1,095

1,130

1,780

2,385

0.3

0.5

1.0

1.5

Page 88

E 2,000 2,400 2.0

The risk free rate is 8% and the expected return on an average market portfolio is 15%.

Required:

i) Using the Capital Asset Pricing Model (CAPM), show the projects that are- acceptable.

ii) Galaxy limited beta factor

iii) Show the projects that would be accepted and rejected if they were discounted at the

firm's1 cost of capital.

Highlight those projects where an incorrect decision would be made.

Solution

Project RP = RF + [RM – RF] 𝜷 ER ER Advice

A

B

C

D

E

8 + [15 – 8] ×0.3 10.1

8 + [15 – 8] ×0.5 11.1

8 + [15 – 8] × 1.0 15.0

8 + [15 – 8] × 1.5 18.5

8 + [15 – 8] × 2.0 22.0

1095−1000

1000× 100

9.5%

13%

18.7%

19.25%

20%

Reject

Accept

Accept

Accept

Reject

Galaxy Limited’s Beta factor

RF + (RM – RF)𝛽 = cost of capital

8 + (15 – 8) x = 18.5

7𝑥

7 =

18.5−8

7

x = 1.5

Project

DF

Cash flows

Sh. 000

Initial Outlays

NPV

Advice

A

B

C

D

E

0.8439

0.8439

0.8439

0.8439

0.8439

1095

1130

1780

2385

2400

(1,000) =

(1,000) =

(1,500) =

(2,000) =

(2000) =

(75.9295)

(46.393)

2.142

12.7015

25.36

Reject

Reject

Accept

Accept

Accept

Illustration 3

Mr. Akili Mingi holds the following portfolio of four risky assets and a deposit in a risk free asset.

The table below shows the respective portfolio weightings and the current returns on the assets,

together with their beta coefficients.

Asset Weighting (%) Current returns (%) Beta coefficient

A

B

20

10

12.0

18.0

1.5

2.0

Accept projects whose expected

returns are higher than their CAPM

returns

Page 89

C

D

Risk-free asset

15

25

30

14.0

5.0

5.0

1.2

0.9

0.0

The overall return on the market portfolio of risky assets is 11% and this is expected to continue for

the foreseeable future.

Required:

(i)The portfolio current return and the portfolio beta. Briefly comment on these two measure

(ii)Determine the assets which are inefficient, efficient or super efficient.

(iii)In view of your answer in (a) (ii) above, predict how the future asset values and, hence, their rates

of return would behave as the market moves towards full equilibrium

Solution

(i) Portfolio Return = ԐReturns of securities x weights of securities

12 x 0.2 + 18 x 0.1 + 14 x 0.15 + 8 x 0.25 + 5 x 0.3

= 9.8%

Portfolio Beta = ԐBeta of securities x weights of securities

1.5 x 0.2 + 2.0 x 0.1 + 1.2 x 0.15 + 0.9 x 0.25 + 0.0 x 0.3

= 0.905

The investor is relatively risk neutral

The systematic risk of the portfolio as measured by the beta factor is lower than the market portfolio.

This indicates that the returns on assets that comprise the portfolio are less sensitive to changes in

returns of the market portfolio.

(ii) Efficient assets lie on security market line (SML) They are correctively valued.

Super-efficient assets offer more than what the Beta values warrant i.e. they are undervalued.

Inefficient assets offer less than what the better values warrant i.e. they are overvalued.

Asset

CAPM

Rj = RF + [RM – RF] 𝜷 Current Return

A

B

C

D

5 + [11 – 5] 1.5 = 14%

5 + [11 – 5] 2.0 = 17%

5 + [11 – 5] 1.2 12.2%

5 + [11 – 5] 0.9 10.4%

12%

18%

14.0%

8.0%

Inefficient (Alpha = 12% - 14% = -2%

Super-efficient (α = 18% - 17% = 1%)

Super-efficient (α = 14% - 12.2% = 1.8%

Inefficient (8 – 10.4% = -1.2%

(iii) Super-efficient assets offer super normal returns hence they are very attractive. Investors will

therefore buy the super-efficient securities which are normally undervalued and will sell

inefficient securities which are normally overvalued.

Securities with negative alpha values are inefficient

Page 90

This will adjust their respective prices until each asset offers a return consistent to its beta factor.

Prices of securities B and C are expected to increase while the prices of securities A and D would

reduce.

b) Optimal weights

Wx = 𝛿2𝑦−𝐶𝑂𝑉 𝑥,𝑦

𝛿2𝑥+ 𝛿2𝑦−2𝐶𝑂𝑉 𝑥,𝑦

= 297.6− −54

10+297.6−2 x−54

351.6

415.6 = 84.6%

Wy = 100% - 84.6% = 15.4%

Difference between CAPM and Portfolio Theory

CAPM Portfolio Theory

It deals with systematic risk

It measures risk using beta factor

Beta of the market is always equal to 1

It uses SML i.e. RF + (RM – RF)𝛽

Beta of portfolio is a weighted average

It deals with total risk

It measures the total risk by of 𝛿

Beta of the market portfolio is not always

equal to 1

Uses CML i.e. RF + [𝑅𝑀−𝑅𝐹

𝛿𝑀] 𝛿𝑃

𝛿𝑃 is not always a weighted average unless

ℓ= 1

2.3. SHORTCOMINGS OF THE CAPITAL ASSET PRICING MODEL

CAPM has several weaknesses e.g.

c. It is based on some unrealistic assumptions such as:

i) Existence of Risk-free assets

ii) All assets being perfectly divisible and marketable (human capital is not divisible)

iii) Existence of homogeneous expectations about the expected returns

iv) Asset returns are normally distributed.

b. CAPM is a single period model—it looks at the end of the year return.

c. CAPM cannot be empirically tested because we cannot test investors’ expectations.

d. CAPM assumes that a security's required rate of return is based on only one factor (the stock

market—beta). However, other factors such as relative sensitivity to inflation and dividend payout,

may influence a security's return relative to those of other securities.

The Arbitrage pricing theory is designed to help overcome these weaknesses.

2.4. ARBITRAGE PRICING MODEL/THEORY (APM/APT)

CAPM is a single factor model which is based on unrealistic assumptions/weaknesses.

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To counter these weaknesses APM was developed which was based on the argument that a

number of factors will influence the returns of a security such as:

Interest Rates, Inflation, Exchange Rate Differences, World Prices, etc.

Arbitrage pricing mode is based on the following assumptions;

i) That capital markets are efficient and perfectly competitive.

ii) Returns of a security are generated through continuous trading of such securities.

iii) Investors prefer more wealth to less i.e. it ignores the concept of indifference curves in

measuring investors utility.

Under APM Rj = RF + [RM1 – RF] 𝛽1 + [RM2 – RF] 𝛽2 + ….. [RMn – RF] 𝛽𝑛

Rj = Return on security jRmn, Rm2, Rm1 – Expected market return of different components of the

systematic riskB1, B2, Bn = Systematic risk of different components in the market.

ARBITRAGE PRICING THEORY BENEFITS

APT model is a multi-factor model. So, the expected return is calculated taking into account various

factors and their sensitivities that might affect the stock price movement. Thus, it allows selection of

factors that affect the stock price largely and specifically.

APT model is based on arbitrage free pricing or market equilibrium assumptions, which to a certain

extent result in a fair expectation of the rate of return on the risky asset.

APT based multi-factor model places emphasis on the covariance between asset returns and

exogenous factors, unlike CAPM. CAPM places emphasis on the covariance between asset returns

and endogenous factors.

APT model works better in multi-period cases as against CAPM, which is suitable for single period

cases only.

APT can be applied to the cost of capital and capital budgeting decisions.

The APT model does not require any assumption about the empirical distribution of the asset returns,

unlike CAPM, which assumes that stock returns follow a normal distribution and thus APT a less

restrictive model.

ARBITRAGE PRICING THEORY LIMITATIONS

The model requires short listing of factors that influence the stock under consideration. Finding and

listing all factors can be a difficult task and runs a risk of some or the other factor being ignored. In

addition, the risk of accidental correlations may exist which may cause a factor to become substantial

impact provider or vice versa.

The expected returns for each of these factors will have to be arrived at, which depending on the

nature of the factor, may or may not be easily available always.

The model requires calculating sensitivities of each factor, which again can be an arduous task and

may not be practically feasible.

The factors that affect the stock price for a particular stock may change over a period. Moreover, the

sensitivities associated may also undergo shifts, which need to be continuously monitored making it

very difficult to calculate and maintain.

Conclusion

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Arbitrage Pricing Theory-based models are built on the principle of capital market efficiency and aim

to provide decision makers and participants with estimates of required rate of return on the risky

assets. The required rate of return arrived using the APT model can be used to evaluate, if the stocks

are over-priced or under-priced. Empirical tests conducted in the past have resulted from APT as a

superior model over CAPM in many cases. However, in several cases, it has arrived at similar results

as CAPM model, which is relatively simpler in use.

Difference between CAPM and APM

CAPM APM/APT

It is a single factor model

i.e. RF + (RM – RF)𝛽

It is a single period model which cannot

be extended beyond 1 period

It assumes normal distribution of security

by use of linear equation

Market portfolio should be well

diversified since it influences the returns

of a security.

It uses indifference curves to measure

utility of individual investors hence it is

more theoretical.

It is considered to be less practical by

considering only one factor in analysing

the systematic risk i.e. market portfolio.

It is a multi-factor mode i.e. Rj = RF + [RM1 –

RF] 𝛽1 + …..+ [RMn – RF] 𝛽𝑛

It is a multi-period mode which can be extended

over several periods in future.

It ignores normal distribution of security return

i.e. security returns are obtained through

continuous trading.

Market portfolio need not to be well diversified

since it has no special role to play.

It ignores indifference curves in measuring

utility of individual investors i.e. it is more

practical.

It is considered to be more realistic since it

considers all factors that lead to systematic risk.

Illustration

An investor is considering investing in the stocks of three companies. A Ltd, B Ltd and C

Ltd. The following information relates to the stocks of the three companies:

Sensitivity of stock’s returns to changes in:

Company Market index Inflation rate Economic growth rate

A Ltd

B Ltd

C Ltd

1.50

0.90

1.10

0.10

0.10

-0.43

0.56

0.60

0.86

During the year 2014, it is expected that the market index will increase in performance by 2.5% up

from its current 5%. The risk free rate of return in the market will be 6% on average and the inflation

and economic growth rates will be 10% and 5.6% respectively.

Required:

(i) Expected returns for the three stocks in year 2014 using the capital asset pricing model

(CAPM)

(ii) Expected returns for the three stocks in year 2014 using the arbitrage pricing theory (APT)

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ADVANCED FINANCIAL MANA

(iii) State the reason why an investor would get different return estimates in (b)(i) and (b)(ii)

above.

Solution

(i) Expected returns of the stocks using CAPM

ERj = Rf + (Rm – Rf) 𝛽𝑖

ERA = 6 + (7.5 – 6) 1.5 = 8.25%

ERB = 6 + (7.5 – 6) 0.90 = 7.35%

ERC = 6 + (7.5 – 6) 1.1 = 7.65%

Rf is the Risk free rate; Rm is the Return on the market;𝛽𝑖 is the Sensitivity of the asset’s return which

is represented by Beta.

(ii)Expected returns of the stocks using the arbitrage pricing theory (APT)

HINT;

APT is a multiples factor model.

APT returns = Rf + (Ri – Rf) 𝛽𝑖 + (RE – Rf) 𝛽𝑒

ERA = 6 + (10 – 6) -0.1 + (5.6 – 6) 0.56 + (7.5-6)1.5 = 7.626%

ERB = 6 + (10 – 6) 0.1+ (5.6 – 6) 0.6 + (7.5-6)0.9 = 7.51%

ERC = 6 + (10 – 6) -0.43 + (5.6 – 6) 0.86 + (7.5-6)1.1 = 5.59%

E(R1) is the assets’ expected; 𝛽𝑖 is the Sensitivity of the asset’s return to a particular factor rate of

return.

(iii) Why an investor will get different return estimates

CAPM is a single factor model that assumes that the only factors that affects portfolio returns is the

market factor (market returns) whereas APT brings on board many factors. These differences in terms

of variables used wil yield different results.

The Fama-French Model

One of the critiques on CAPM was that it seemed to fail to explain why small cap stocks tended to

outperform large cap stocks. Eugene Fama and Kenneth French researched this issue and introduced a

new model, an extended form of CAPM, called Fama-French Model (FFM). FFM includes the

following factors:

RMRF – Stands for the difference between market return and risk free return, as in CAPM.

SMB – Stands for small minus big, thus is a size factor. It is estimated as the difference between

average return on three small-cap portfolios minus average return on three large-cap portfolios.

Thus, SMB represents a return premium over the large cap stocks.

HML – Stands for high minus low, the average return on two high book-to-market portfolios

minus the average return on two low book-to-market portfolios. This factor accounts for the value

NOTE:Ri = Rf + Beta (market) x RMRF + Beta

(size) x SMB + Beta (value) x HML

Page 94

return premium and the effect of value stocks (high book-to-market) and growth stocks (low

book-to market).

Where Beta (market), Beta (size) and Beta (value) are sensitivities to RMRF, SMB and HML,

respectively.

Just as in the CAPM, sensitivity factors (betas) in Fama-French can be estimated based on historical

data by linearly regressing asset excess returns with respective premiums (multivariable regression).

The application of the Fama-French model in practice is complex and therefore cannot be

illustrated within this context.

The multivariable linear regression will give us historical betas, which can be a starting point for us to

estimate ex-ante betas.

Pastor and Stambaugh suggested adding a new factor, liquidity premium, to FFM, thus extending

the model. The idea behind this was that investors demand additional premium for holding illiquid

assets. Thus, the formula becomes:

Ri = Rf + Beta (market) *RMRF + Beta(size)*SMB + Beta (value) * HML + Beta (liq)*LIQ

*Where LIQ is the premium for liquidity.

2.5. PORTFOLIO PERFORMANCE MEASURES In measuring the performance of a portfolio, the following methods can be applied;

i) Treynor’s measure

ii) Sharpe’s measure

iii) Jensen’s measure

iv) Appraisal Information Ratio

v) Modigliani and Modigliani

Treynor’s Measure

Treynor was the first scholar to come up with a composite measure of portfolio performance. This

measure is based on the background of CAPM and therefore the Assumptions and limitations of

CAPM also applicable to the Treynor’s measure of portfolio performance.

The Treynor reward to volatility model (sometimes called the reward-to-volatility ratio or Treynor

measure, named after Jack L. Treynor, is a measurement of the returns earned in excess of that which

could have been earned on an investment that has no diversifiable risk (e.g., Treasury bills or a

completely diversified portfolio), per each unit of market risk assumed.

The Treynor ratio relates excess return over the risk-free rate to the additional risk taken; however,

systematic risk is used instead of total risk. The higher the Treynor ratio, the better the performance of

the portfolio under analysis.

It measures the portfolio performance by incorporating systematic risk as measured by the beta factor

i.e.

Tj = 𝑅𝑗−𝑅𝐹

𝛽

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Tj = Treynor’s measure of security

Rj = E Returns of security on the market

RF = Risk Free Rate of Return

𝛽𝑗 = Systematic Risk/Beta factor of security

This measure is compared with a similar measure of the market portfolio for analysis purposes i.e.

Tm = 𝑅𝑚−𝑅𝐹

𝛽𝑚

Recall:

𝛽𝑚 = 1

[Rm – RF] = Risk Premium

Tm = Risk Premium

Analysis

If Tj is more than the Tm, it indicates a superior performance

If Tj is less than Tm, (Tj< Tm) it indicates an inferior performance

If Tj is equal to Tm it indicates an efficient performance

Sharpe’s Measure

The Sharpe ratio is defined as the risk premium of the portfolio per unit of total risk in the portfolio.

Risk premium calculated by subtracting risk-free returns from the portfolio returns. The risk-free

returns are measured as the risk-free interest rate of Treasury bonds.

It measures portfolio performance by use of total risk as measured by 𝛿 i.e.

𝛿𝑗 = 𝑅𝑗−𝑅𝐹

𝛿𝑗

Sj = Sharpe’s measure of security

𝛿𝑗 = Standard deviation of security j

Rj = E returns of security in the market

RF = risk free rate of return

For evaluation purposes, a Sharpe’s measure of a security is compared with a similar or measure of

the market portfolio

Sm = 𝑅𝑚−𝑅𝐹

𝛿𝑚

Analysis

If Sj>Sm, it indicates a superior performance

If Sj<Sm, it indicates an inferior performance

If Sj = Sm, it indicates an efficient performance

Jensen’s Measure

the Jensen's measure is a risk-adjusted performance measure that represents the average return on

a portfolio or investment, above or below that predicted by the capital asset pricing model (CAPM),

given the portfolio's or investment's beta and the average market return. This metric is also commonly

referred to as Jensen's alpha, or simply alpha.

With an assumption that capital markets are efficient and perfect, CAPM becomes accurate in

estimating returns of a portfolio.

Jensen’s measure uses Alpha values (𝛼) in measuring portfolio performance.

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𝛼 = ERP – Required return from portfolio

Where: RP = CAPM

RP = RF + [RM – RF] 𝛽

Alpha 𝛼 = ERP – RF + [RM – RF] 𝛽

Analysis

If 𝛼 value is greater than 0, it indicates a superior performance

If 𝛼< 0, it indicates an inferior performance

If 𝛼 = 0, it indicates an efficient performance

Appraisal /Information Ratio

The appraisal ratio is a ratio used to measure the quality of a fund manager's investment-picking

ability. It compares the fund's alpha to the portfolio's unsystematic risk or residual standard deviation.

The fund's alpha is the amount of excess return the manager has earned over the benchmark of the

fund. It is the portion of the return that the portfolio manager's active management is responsible for.

The ratio shows how many units of active return the manager is producing per unit of risk.

It measures portfolio average return in excess of comparative benchmark portfolio.

IR/AR = 𝐸𝑅𝑃−𝐸𝑅𝐵

𝛿𝐸𝑅

ERP = Expected Portfolio Return

ERB = Expected Benchmark Portfolio

𝛿ER = 𝛿𝑃 − 𝛿𝐵

Analysis

The higher the ratio, the better the performance

Illustration 1

The risk and return characteristics of two assets are as shown below:

Asset A B

Expected return

Risk (standard deviation)

12%

3%

20%

7%

Uchumi investment Company plans to invest 80% of its available funds in asset A and 20% in asset

B. The board of directors of the company believe that the correlation coefficient between the returns

of these assets is +0.1.

Required:

(i) The expected return from the proposed portfolio asset A and asset B.

(ii) The risk of the portfolio.

Page 97

(iii) Comment on your calculations in part (c) (ii) above in the context of the risk-reducing effects

of diversification.

(iv) Suppose the correlation coefficient between the returns of asset A and asset B was -1.0.

Demonstrate how Uchumi Investment Company could invest its funds in order to obtain a

zero-risk portfolio.

Solution

(i) The expected return from the proposed portfolio asset A and asset B.

ER = RAWA + RBWB

12% ×80% + 20% × 20% = 13.6%

(ii) The risk of the portfolio.

Risk of the portfolio = 𝛿P = √𝛿𝐴2𝑊𝐴

2 + 𝛿𝐵2𝑊𝐵

2 + 2𝑊𝐴𝑊𝐴𝐶𝑜𝐴, 𝐵

COV A,B = ℓA,B x 𝛿A x 𝛿B

0.1 x 0.3 x 7 = 2.1%

= √32 × 0.82 + 72 × 0.22 + 2 × 0.8 × 0.2 × 2.1

= 2.90%

(iii)Comment on your calculations in part (c) (ii) above in the context of the risk-reducing effects

of diversification.

Through diversification the overall risk of the portfolio reduces to 2.90% as compared to a higher risk

of 3% for assets A and 7% for asset B.

Weighted σP = WAσA + WBσB = 0.8 x 3% + 0.2 x 7% = 3.8%

The risk has reduced from 3.8% to 2.9%

(iv)

To calculate the proportions investment of types of assets in a portfolio we use the minimum weight

formula i.e.

Wx = 𝛿2𝑦−𝐶𝑂𝑉 𝑥,𝑦

𝛿2𝑥 𝛿2𝑦−2𝐶𝑂𝑉 𝑋,𝑌

Wy = 1 – Wx

WA = 𝛿2𝑦−𝐶𝑂𝑉 𝐴,𝐵

𝛿2𝐴 + 𝛿2𝐵 −2𝐶𝑂𝑉 𝐴,𝐵

COV A,B = ℓ A,B x 𝛿A x 𝛿B

-0.1 x 3 x 7 = -21

72−−21

32+ 72−2∗−21 =

70

100

PAB = Correlation

Coefficient between A & B

Page 98

WA = 70%

WB = 100%-70%=30%

To obtain a zero risk portfolio, 70% of Uchumi’s investment should come from asset A and 30% from

asset B.

Illustration 2

The following information relates to portfolios P and N:

Average return

Beta

Standard deviation

Non-systematic risk

Portfolio P

35%

1.25

42%

18%

Portfolio N

28%

1.00

30%

10%

Assume that the risk free rate is 6% and the average market return is 15%.

Required:

(i) Sharpe’s performance measure for portfolios P and N.

(ii) Treynor’s performance measure for portfolios P and N.

(iii) Jensen’s performance measure for portfolios P and N.

(iv) The appraisal ratio for portfolios P and N.

Solution

(i) Sj = 𝑅𝑠−𝑅𝑓

𝛿𝑠

Portfolio P 35−6

42 = 0.69

Portfolio N 28−6

30 = 0.73

(ii) Tj = 𝑅𝑇−𝑅𝐹

𝐵𝑇

Portfolio P 35−6

1.25 = 23.2

Portfolio N 28−6

1.0 = 22

(iii) Jensen’s portfolio measure

P

αP = ERM – [RF + Bj (ERM – RF)

= 35 – [6% + 1.25 (15% - 6%)

Page 99

= 35 – 17.25= 17.75

N

αN = 28% - [6% + 1(15% - 6%)

= 28% - 15% = 13%

(iv) Appraisal ratio

= α (Alpha)

Non-systematic risk (δ)

P

= 17.75%

18% = 0.99

N

= 13%

10% = 1.3

5. Modigliani and Modigliani M2

Definition: In simple words, it measures the returns of an investment portfolio for risk taken relative

to some benchmark portfolio. Popularly known as Modigliani Risk Adjustment Performance Measure

or M2, it was developed by Nobel Prize winner Franco Modigliani and his granddaughter Leah

Modigliani in the year 1997.

Description: Franco Modigliani and Leah Modigliani believed that an ordinary investor would find it

easier to understand the Modigliani measure compared to Sharpe ratio. The reason behind this was

that their measure is expressed in percentage points. It shows how well the investor is rewarded for

taking a certain amount of risk, relative to the benchmark and the risk free rate.

A fund, which has taken same risk as that of the benchmark but generates better returns, will have

superior risk return trade-off as compared a fund that has taken a significantly higher risk, but gives

almost similar returns as that of the benchmark.

Computation Modigliani and Modigliani M2

Step 1, we need to determine Sharpe ratio as follows

𝑆𝑅 =𝑟𝑝 − 𝑟𝑓

δp

Step 2 consist of multiplying the Sharpe ratio with annualized standard deviation of the bench mark

Step 3 add the risk free rate of return to Sharpe ratio

M2= SR ×Standard deviation of the benchmark+ rf

M2=𝑟𝑝−𝑟𝑓

δp × δBench + rf

Alternatively

M2=𝑟𝑝−𝑟𝑓

βp × βBench + rf

Illustration 1

The following information relates to the performance of six portfolios over a seven-year period:

Portfolio Average annual

returns (%)

Standard deviation of the

average annual returns (%)

Correlation with

market returns

P

Q

R

18.6

14.8

15.1

27.0

18.0

8.0

0.81

0.65

0.98

Page 100

S

T

U

Market return

Risk-free rate

22.0

-9.0

26.5

12.0

9.0

21.2

4.0

19.3

12.0

0.75

0.45

0.63

Required:

Rank the performance of the above portfolios using:

(i) Sharpe’s method

(ii) Treynor’s method

(c)Compare the rankings using the two methods in (b) above and explain two reasons behind the

differences.

Solution

(i) Sharpe’s method:

HINT;

Sharpe’s model uses standard deviation as a basis of evaluating securities.

Note;

Generally the higher the Sharpe coefficient is the better a security.

S= (Rp – Rf) ÷ 𝛿p

(ii) Treynor’s method:

HINT

Treynors model uses Beta factor to evaluate the performance of securities.

NOTE

First compute Beta factors of securities

Tp= Rp - Rf ÷ 𝛽D

𝛽s =𝐶𝑂𝑉𝑆𝑀

𝛿𝑚2

COVsm =rsm 𝛿𝑠𝛿𝑚

P = (18.6 – 9) ÷ 27 =0.3555

Q = (14.8 – 9) ÷ 18 =0.3222

R = (15.1 – 9) ÷ 8 =0.7625

S = (22 – 9) ÷ 21.2 =0.6132

T = (-9.0 – 9) ÷ 4 =-4.5

U = (26.5 – 9) ÷ 19.3=0.9067

Rank

4

5

2

3

6

1

Page 101

Therefore Bs = 𝑟𝑠𝑚𝛿𝑠𝛿𝑠

𝛿𝑚𝑥𝛿𝑚

𝛽s = 𝑟𝑠𝑚𝛿𝑠

𝛿𝑚

𝜷𝒔= 𝒓𝒔𝒎𝝈𝒎

𝝈𝒎

(Rs - Rf) ÷ 𝜷s

Rank

P = 27 ×0.81 ÷ 12 = 1.823

Q = 18 × 0.65 ÷ 12 = 0.975

R = 8 × 0.98 ÷ 12 = 0.653

S = 21.2 ×0.75 ÷ 12 = 1.325

T = 4.00 × 0.45 ÷ 12 = 0.1

U = 19.3×0.63 ÷ 12 = 1.013

(18.6 – 9) ÷ 1.823= 5.266

(14.8 – 9) ÷ 0.975= 5.95

(15.1 – 9) ÷ 0.653= 9.342

(22 – 9) ÷ 1.325= 9.811

(-9.0 – 9) ÷ 0.15= -120

(26.5 – 9) ÷ 1.013= 17.27

4

5

2

3

6

1

(c)Reasons for the differences in ranking:

Sharpe’s index considers only the standard deviation and correlation whereas Treynor considers

market return standard deviation and correlation.

Sharpes consider total risk where as Treynors considers systematic risk only.

Illustration 2

The following information relates to portfolios P and N:

Average return

Beta

Standard deviation

Non-systematic risk

Portfolio P

35%

1.25

42%

18%

Portfolio N

28%

1.00

30%

10%

Assume that the risk free rate is 6% and the average market return is 15%.

Required:

(i)Sharpe’s performance measure for portfolios P and N.

(ii)Treynor’s performance measure for portfolios P and N.

(iii)Jensen’s performance measure for portfolios P and N.

(iv)The appraisal ratio for portfolios P and N.

Solution

(i)Sharpe’s performance measure

Sj = 𝑅𝑠−𝑅𝑓

𝛿𝑠

Portfolio P 35−6

42 = 0.69

Portfolio N 28−6

30 = 0.73

(ii)Treynor’s performance measure

Tj = 𝑅𝑇−𝑅𝐹

𝐵𝑇

Page 102

Portfolio P 35−6

1.25 = 23.2

Portfolio N 28−6

1.0 = 22

(iii)Jensen’s performance measure

Jensen’s portfolio measure

P

αP = ERM – [RF + Bj (ERM – RF)

= 35 – [6% + 1.25 (15% - 6%)

= 35 – 17.25

= 17.75

N

αN = 28% - [6% + 1(15% - 6%)

= 28% - 15%

= 13%

(iv)Jensen’s performance measure

Appraisal ratio

= α (Alpha)

Non-systematic risk (δ)

P

= 17.75%

18% = 0.99

N

= 13%

10% = 1.3

EXAM PRACTISE QUESTIONS

Question 1

The risk and return characteristics of two assets are as shown below:

Asset A B

Expected return

Risk (standard deviation)

12%

3%

20%

7%

Uchumi investment Company plans to invest 80% of its available funds in asset A and 20% in asset

B. The board of directors of the company believe that the correlation coefficient between the returns

of these assets is +0.1.

Required:

(i)The expected return from the proposed portfolio asset A and asset B.

(ii)The risk of the portfolio.

(iii)Comment on your calculations in part (c) (ii) above in the context of the risk-reducing effects of

diversification.

(iv)Suppose the correlation coefficient between the returns of asset A and asset B was -1.0.

Demonstrate how Uchumi Investment Company could invest its funds in order to obtain a zero-risk

portfolio.

Page 103

Solution

(i)The expected return from the proposed ER = RAWA + RBWB 12% × 80% + 20% × 20%= 13.6%

(ii)The risk of the portfolio

Risk of the portfolio = 𝛿P = √𝛿𝐴2𝑊𝐴

2 + 𝛿𝐵2𝑊𝐵

2 + 2𝑊𝐴𝑊𝐴𝐶𝑜𝐴, 𝐵

COV A,B = ℓA,B x 𝛿A x 𝛿B

0.1 x 0.3 x 7 = 2.1%

= √32 × 0.82 + 72 × 0.22 + 2 × 0.8 × 0.2 × 2.1= 2.90%

(iii)Comment on your calculations in part (c) (ii) above

Through diversification the overall risk of the portfolio reduces to 2.90% as compared to a

higher risk of 3% for assets A and 7% for asset B.

Weighted σP = WAσA + WBσB = 0.8 x 3% + 0.2 x 7% = 3.8%

The risk has reduced from 3.8% to 2.9%

(iv)How Uchumi Investment Company could invest its funds in order to obtain a zero-risk

portfolio. To calculate the proportions investment of types of assets in a portfolio we use the minimum

weight formula i.e.

Wx = 𝛿2𝑦−𝐶𝑂𝑉 𝑥,𝑦

𝛿2𝑥 𝛿2𝑦−2𝐶𝑂𝑉 𝑋,𝑌

Wy = 1 – Wx

WA = 𝛿2𝑦−𝐶𝑂𝑉 𝐴,𝐵

𝛿2𝐴 + 𝛿2𝐵 −2𝐶𝑂𝑉 𝐴,𝐵

COV A,B = ℓ A,B x 𝛿A x 𝛿B

-0.1 x 3 x 7 = -21

72−−21

32+ 72−2∗−21 =

70

100

WA = 70%

WB = 100%-70%=30%

To obtain a zero risk portfolio, 70% of Uchumi’s investment should come from asset A and 30% from

asset B.

Question 2

PAB = Correlation

Coefficient between A & B

Page 104

(a) Biashara Ltd. wishes to invest in stocks M and N in two different industries. The following

information relates to the two stocks:

Stock M Stock N

Expected return

Standard Deviation

Beta coefficient

Amount of money invested (Sh.)

18

8

1.80

1,200,000

16

6

1.50

800,000

Required;-

(i)The expected portfolio return.

(ii)Explain the effect on the risk if the returns of stock M and N were perfectly positively correlated.

Include suitable calculations.

(b) Mapeni Ltd’s investment fund comprises major projects. The details of the projects are as

follows:

Project Market value

of the fund (%)

Expected

return (%)

Standard

deviation (%)

Coefficient of correlation with

the market

1

2

3

4

28

17

31

24

10

18

15

13

15

20

14

18

0.55

0.75

0.84

0.62

The risk-free rate is 5% and the market return is 14%. The standard deviation of the market return is

13%.

Required:

(i)The beta coefficient of the investment fund.

(ii)By comparing the expected return and the required return, advise whether Mapeni Ltd should

change the composition of its portfolio.

Solution

HINT;

A portfolio is a combination of many investments.

(a)

(i)Expected portfolio return:

Note;

Use the investment amount of securities to ascertain the weights in the portfolios

Expected returns Investment

Sh.

Weights

M 18 1,200,000 1,200,000

2000000=0.6

N 16 800,000 800,000

2,000,000=0.4

Page 105

2,000,000

E (Rp) = (WM ERM+ (WNE(RN) = (0.6×18) + (0.4×16) = 17.2%

(ii)If the returns of stocks are perfectly positively correlated, the returns are expected to move in the

same direction. However, more importantly, a perfect positive correlation is interpreted to mean that

there is 0% risk reduction through building of such a portfolio. Therefore, building a portfolio whose

stock returns are perfectly positively correlated renders portfolio building meaningless.

In order to justify this argument, we need to determine the 0% risk reduction through portfolio

building as follows:

0% risk reduction = 𝑊𝑒𝑖𝑔ℎ𝑡𝑒𝑑𝛿𝑝–𝐴𝑐𝑡𝑢𝑎𝑙𝛿𝑝

Weighted dp

Weighted portfolio risk (𝛿p)

= (wm𝛿m) + (wn𝛿n)= (0.6×8) + (0.4×6) = 7.2

Actual portfolio risk (𝛿p)

Note;

First determine COVMN =rMN𝛿M𝛿N

rMN = correlation coefficient between M and N= 1 since the securities are perfectly positively related.

HINT;

Perfectly positively correlated securities have a correlation coefficient of 1

Therefore COVMN =1×8×6 =48

Actual 𝜹p

𝛿2p =Wm2𝛿M2 + Wn2𝛿n2 + 2COVMNWMWN

=0.62 × 82 + 0.42 × 62 + 2×48×0.6×0.4=51.84

𝛿p=√51.84 =7.2

% risk reduced=𝑤𝛿𝑝−𝐴𝛿𝑃

𝑊𝛿𝑃=

7.2−7.2

7.2 × 100= 0%

Where;-

W𝛿p = weighted standard deviation of a portfolio.

A𝛿p = Atual

(b)

(i)The investment fund’s Beta coefficient:

Beta of an individual asset (𝛽j)

Page 106

BJ =𝐶𝑂𝑉𝑗,𝑚

𝛿𝑚2 = 𝑟𝑗𝑚×𝛿𝑗×𝛿𝑚

𝛿𝑚×𝛿𝑚

Project 1

B1 = 𝑟𝑗𝑚.𝛿𝑗

𝛿𝑚=

0.55 ×15

13 =0.635

Project 2

B2 = =𝑟2𝑚 .𝛿2

𝛿𝑚 =

0.75 ×20

13 =1.154

Project 3

B3 = =𝑟3,𝑚 .𝛿3

𝛿𝑚 =

0.84 ×14

13 =0.905

Project 4

B3 = =𝑟4,𝑚 .𝛿4

𝛿𝑚 =

0.62×18

13 =0.86

Investment fund’s beta Coefficient (Bp)

HINT

For w=weights, use the market values of funds (%). E.g. project 1 =28%, W1 =0.28

Bp = (W1𝛽1) + (W2𝛽2) + (W3𝛽3) + (W4𝛽4)

= [0.28×0.635] + [0.17×1.154] + [0.31×0.905] + [0.24×0.86] = 0.861

E (portfolio return) = E Rp

E Rp = [(0.28 × 10) + (0.17×18) + (0.31×15) + (0.24×13)] = 13.63%

Required portfolio return (Rp)

HINT;

Use the CAPM to determine the required rate of return of each project.

CAPMRs= Rf + (Erm – Rf) 𝛽s

Project Required return % Weights

1

2

3

4

R1 = 5 + (14.5)0.635

R2 = 5 + (14-5)1.154

R3 = 5 + (14-5)0.905

R4 = 5 + (14-5)0.86

10.72

15.39

13.15

12.74

0.28

0.17

0.31

0.24

Page 107

Rp = [0.28×10.72) + (0.17×15.39) + (0.31×13.15) + (0.24×12.74)] = 12.75%

Portfolio is profitable because is expected rate of return is greater than the required rate of return.

Thus, there is no need to change the composition of this portfolio.

The Jensens Alpha value= ERp –CAPM Rp

𝛼p= ERp –CAPM Rp= 13.63% -12.75% =0.88%

= it is positive implying that the portfolio is profitable i.e. it’s undervalued.

Illustration 3

a) In most cases, the assumption is that investors are risk-averse, that is, they like returns and dislike

risk.

With reference to the above statement, explain why it is argued that only systematic risk and not

total risk is important.

b) In the context of portfolio theory, explain the meaning of "beta coefficient".

c) The following data have been provided with respect to three shares traded on the Nairobi

Securities Exchange (NSE):

Share A Share B Share C

Risk-free rate of return

Beta coefficient

Return on the NSE index

12%

1.340

0.185

12%

1.000

0.185

12%

0.750

0.185

Required:

i) Interpret the beta coefficients of shares A, B and C.

ii) Using the capital asset pricing model (CAPM), compute the expected return on shares A, B

and C.

d) The following information relates to portfolios P and N:

Portfolio P Portfolio N

Average return

Beta

Standard deviation

Non-systematic risk

35%

1.25

42%

18%

28%

1.00

30%

10%

Assume that the risk free rate is 6% and the average market return is 15%.

Required:

i) Sharpe's performance measure for portfolios P and N.

ii) Treynor's performance measure for portfolios P and N.

iii) Jensen's performance measure for portfolios P and N.

iv) The appraisal ratio for portfolios P and N.

Solution

(a)Explain why it is argued that only systematic risk and not total risk is important.

Page 108

Total risk is an amalgamation of systematic risk and the unsystematic risk. However, as investors

invest in more than one company, that is portfolio, the unsystematic risk is eliminated and therefore,

the only relevant risk component that will determine the return is the systematic risk and that is why

we consider systematic risk only.

CAPM is the model that works with systematic risk to evaluate portfolio security returns.

Systematic risk is measured using the beta risk (B)

CAPMRs= Rf + (Erm – Rf) 𝛽s

Diversification principle

(b) The meaning of "beta coefficient".

Beta coefficient (𝛽𝑗) is a measure of the sensitivity of the returns on a security or a portfolio to

changes in the market portfolio.

𝛽𝑗 = 𝐶𝑂𝑉𝑗𝑚,𝑅𝑚

𝜎𝑚2

𝛽𝑗 Of market portfolio = 1

(c)

(i) Interpretation of the beta coefficient of shares:

𝛽𝑗 of A = 1.340

This implies that if the returns of the market portfolio change by one unit, those of share A change by

1.340. It basically implies the returns of share A are sensitive to the changes in the market portfolio.

𝛽𝑗of B = 1.000

This implies that if the returns on the market portfolio change by a unit, then the returns of share B

also change by one unit. These shares are of comparable risk to the market portfolio.

𝛽𝑗of C = 0.750

Implies that if the returns of the market portfolio change by a unit then those of C change by 0.750. It

implies that the returns of share C are less sensitive to changes in the market portfolio and they are

less risky.

Unsystematic risk

Number of securities

Systematic risk

Risk

Page 109

(ii)Using CAPM, the expected return is given by:

HINT;

CAPM uses the systematic (beta factor) to evaluate security returns.

CAPM Rj = Rf + (Rm – Rf) 𝛽𝑖

CAPM Returns of eachsecurity

Share A

Share B

Share C

=

=

=

12 + (18.5 – 12)1.34 =

12 + (18.5 – 12) 1 =

12 + (18.5 – 12)0.75 =

20.71%

18.5 %

16.875 %

(d)

(i) Sharpe's performance measure for portfolios P and N.

HINT;

Sharpe’s model uses standard deviations to evaluate performance of securities.

Sharpe measure SP = 𝐸𝑅−𝑅𝑓

𝛿𝑝 =

0.35−0.06

0.42= 0.690

SN = 0.28−0.06

0.30= 0.733

(ii)Treynor's performance measure for portfolios P and N.

HINT;

Treynors model uses beta factors to evaluate performance of securities.

Treynor’s measure

TP= 𝐸𝑅𝑝−𝑅𝑓

𝛽𝑝 =

0.35−0.06

1.25= 0.232

TN= 0.28−0.06

1 = 0.22

(iii) Jensen's performance measure for portfolios P and N.

HINT;

First step in evaluation of Jensens values is to determine the returns by use of CAPM model.

CAPM RP=Rf + (Rm – Rf)𝛽𝑗

CAPM RP=0.06 + (0.15 – 0.06) 1.25=0.1725

CAPM RN=0.06 + (0.15 – 0.06) 1 = 0.15

𝛼j =ERj – CAPM Rj

Note

ERj is the expected return = average return

Page 110

ERp =0.35

ERN =0.28

𝛼P= 0.35 – 0.1725= 0.1775=17.75%

𝛼N=0.28 – 0.15= 0.13=13%

(iv) The appraisal ratio/information ratio

Appraisal ratio (AP) = 𝐸𝑥𝑐𝑒𝑠𝑠𝑟𝑒𝑡𝑢𝑟𝑛

𝐸𝑥𝑐𝑒𝑠𝑠𝑠𝑡𝑑𝑑𝑣𝑛

HINT;-

For appraisal Ratio model we must be given a benchmark security.

ERB is the return from benchmark security.

𝛿B= Standard deviation benchmark.

Excess return = ERs – ERB

Excess standard deviation = δs – δB

Appraisal ratio (AP) = 𝐸𝑅𝑠−𝐸𝑅𝐵

δs –δB

HINT;-

If benchmark is not given use that of the market portfolio to ascertain the appraisal ratio.

The 𝛿 of the market portfolio (𝛿m) is not provided and therefore the equation given above is irrelevant.

See this!

AP for portfolio P = 35−15

42 –δm

Alternatively the AP ratio can be ascertained as follows

AP = 𝐸𝑅𝑠−𝐸𝑅𝑚

Non−systematic Risk of the security

AP for portfolio P = 35−15

18 = 1.1

AP for portfolio N = 28−15

10 = 1.3

Page 111

CHAPTER THREE

ADVANCED FINANCING DECISION CHAPTER KEY OJECTIVES

To be able to understand the following;-

1. The nature of financing decision, principle objectives of making financing decision

2. Overview of cost of capital: meaning and relevance of cost of capital: the firm’s overall cost of

capital; weighted average cost of capital (WACC) and weighted marginal cost of capital (WMCC)

; analysis of breakpoints in weighted marginal cost of capital schedule

3. Capital structure theories: nature of capital structure and factors influencing the firm’s capital

structure; traditional theories of capital structure - assumptions of the theories, Net income theory

and Net operating income theory; Franco Modigliani and Merton Miller’s propositions - MM

without taxes, MM with corporation taxes, MM with corporation and personal tax rates and MM

with taxes and financial distress costs; other theories of capital structure; the pecking order theory

and Trade-off theory determination of the firm’s optimal capital structure using the Hamada

model, CAPM and WACC

4. Special topics in financing decision: analysis of operating profit (EBIT)/EPS at point of

indifference in firm’s earnings; establishing the range of operating profit within which each

financing option; leverage and risk; operating leverage and operating risk, financial leverage and

financial risk, combined leverage and total risk; quantifying leverage using the degree of

operating leverage, degree of financial leverage and degree of combined leverage

5. Long term financing decisions; bond refinancing decision, lease-buy evaluation and the rights

issues

6. Impact of financing on investment decisions - the concept of adjusted present value (APV)

3.1. NATURE AND SIGNIFICANCE OF FINANCING DECISIONS This refers to the decisions that involve the manner in which the company is expected to raise funds

that are required to finance all profitable projects.

The required amount should always be raised in the most economical way and from the most

economic sources.

In raising the required finances to undertake all profitable projects, the company can utilise common

equity capital, preference share capital or long term debts/debentures.

The cost of raising the required amount is known as the Weighted Average Cost of Capital. This is the

cost of funds utilised to finance the existing projects.

Therefore, financial decisions refer to decisions concerning financial matters of a business concern.

Decisions regarding magnitude of funds to be invested to enable a firm to accomplish its ultimate

goal, kind of assets to be acquired, pattern of capitalization, pattern of distribution of firms, income

and similar other matters are included in financial decisions

The principle objective of making financing decision is to minimize on the cost of borrowing.

Other objectives include

Ensuring that funds are made available within the correct length of time

Ensuring that funds raised are utilised in a more effective manner.

Once the objective have been attained it is possible to achieve the firms objective of wealth

maximization and profit maximum

Page 112

3.2. OVERVIEW OF COST OF CAPITAL

Cost of capital is an integral part of investment decision as it is used to measure the worth of

investment proposal provided by the business concern. It is used as a discount rate in determining the

present value of future cash flows associated with capital projects. Cost of capital is also called as cut-

off rate, target rate, hurdle rate and required rate of return. When the firms are using different sources

of finance, the finance manager must take careful decision with regard to the cost of capital; because

it is closely associated with the value of the firm and the earning capacity of the firm.

Meaning of Cost of Capital

Cost of capital is the rate of return that a firm must earn on its project investments to maintain its

market value and attract funds. Cost of capital is the required rate of return on its investments which

belongs to equity, debt and retained earnings. If a firm fails to earn return at the expected rate, the

market value of the shares will fall and it will result in the reduction of overall wealth of the

shareholders. It is the minimum required rate of return to providers of capital

Significance of cost of capital

Investment Evaluation:

The primary objective of determining the cost of capital is to evaluate a project. Various methods used

in investment decisions require the cost of capital as the cut-off rate. Under net present value method,

profitability index and benefit-cost ratio method the cost of capital is used as the discounting rate to

determine present value of cash flows. Similarly, a project is accepted if its internal rate of return is

higher than its cost of capital. Hence, cost of capital provides a rational mechanism for making the

optimum investment decision.

Designing Debt Policy

The cost of capital influences the financing policy decision, i.e. the proportion of debt and equity in

the capital structure. Optimal capital structure of a firm can maximize the shareholders’ wealth

because an optimal capital structure logically follows the objective of minimization of overall cost of

capital of the firm. Thus while designing the appropriate capital structure of a firm cost of capital is

used as the yardstick to determine its optimality.

Project Appraisal:

The cost of capital is also used to evaluate the acceptability of a project. If the internal rate of return of

a project is more than its cost of capital, the project is considered profitable. The composition of

assets, i.e. fixed and current, is determined by the cost of capital. The composition of assets, which

earns return higher than cost of capital, is accepted.

In measuring top management performance

It is used as a yard stick in measuring top management performance this is done by comparing return

if the actual return is more than the or equal to the minimum required rate of return the management

would be doing best their jobs otherwise they would doing worst their job.

Overall cost of capital

Page 113

The firm's overall cost of capital is based on the weighted average of the costs of equity capital debt

capital and preference shares capital.

The component costs are the cost of equity (Ke) cost of preference shares (Kp), cost of debt (Kd)

Cost of equity

It is the minimum return that should be obtained from the projects that are fully financed by common

equity capital so as to raise cash flows required to pay back to ordinary shareholders. It is therefore

the minimum rate of return required by ordinary shareholders.

Common equity capital is made up of external equity (ordinary shares) and internal equity (Retained

earnings).

The cost of equity in both cases in calculated in the same way except that in case of calculation costs,

such costs would only affect ordinary shares i.e.

Ke (without floatation costs) would be:

Ke = 𝐷𝑜 (1+𝑔)

𝑃𝑜+ 𝑔

Where

Ke – Cost of equity/minimum rate of return required by ordinary shareholders

Do – Current year’s dividend/Dividends just paid/Dividend for the year just ended

g – Annual growth rate in dividend

Po – Current market price of share/intrinsic value per share

In case the dividend doesn’t grow

Ke = 𝐷𝑜

𝑃𝑜−𝐹

Where D1 = Next year’s dividends/Dividends to be paid in a year’s time/Expected dividends

D1 = Do (1 + g)

Ke = 𝐷𝑜 (1+𝑔)

𝑃𝑜−𝐹+ 𝑔 or

𝐷1

𝑃𝑜−𝐹+ 𝑔

In case of floatation costs

Ke (Retained Earnings) = 𝐷𝑜 (1+𝑔)

𝑃𝑜+ 𝑔 or

𝐷1

𝑃𝑜+ 𝑔

Ke (Ordinary shares) = 𝐷𝑜 (1+𝑔)

𝑃𝑜−𝑓+ 𝑔 or

𝐷1

𝑃𝑜−𝑓+ 𝑔

Page 114

Where f = Floatation/issue costs

Illustration

The current market value of ordinary shares of Ujuzi Ltd is Sh. 45. The company has just paid a

dividend of Sh. 5 which grows at annual rate of 8% p.a. ordinary shares attract a floatation cost of 5%

Required;-

Calculate the cost of equity assuming

Solution

Ke (Retained Earnings) = 𝐷1

𝑃𝑜+ 𝑔=

5(1.08)

45 + 0.08 = 0.2 ×100 = 20%

Ke (Ordinary Shares) = 𝐷1

𝑃𝑜−𝑓+ 𝑔=

5(1.08)

45−2.25 + 0.08 = 0.2063×100 = 20.63%

NOTE: Retained earnings do not have floatation costs.

Floatation cost of ordinary shares = 5% x 45 = 2.25

Illustration

A company expects to pay a dividend of Sh. 12 in the coming years on its shares currently selling for

Sh. 60.The past information about the earnings per share for the current year are as analysed below.

The retention ratio is 20%.

Year EPS (Sh.) DPS 80%

2015

2014

2013

2012

2011

2010

14

11

10.5

9.5

8.0

7.0

11.2

8.8

8.4

7.6

6.4

5.6

Note: Retained ratio + Dividend payout ratio = 1

Required

Calculate Ke

Solution

Ke = 𝐷𝑜 (1+𝑔)

𝑃𝑜−𝐹+ 𝑔

D0(1 + g) = D1

Where D1 = Expected dividend = 12

Ke = 12

60 + 0.15

g = √𝑙𝑎𝑡𝑒𝑠𝑡 𝐷.𝑃.𝑆

𝐸𝑎𝑟𝑙𝑖𝑒𝑠𝑡 𝐷.𝑃.𝑆

𝑛 = √

11.2

5.6

5

= 1.15 – 1 = 0.15 = 15%

Page 115

= 0.35 x 100 = 35%

Note: The growth rate of dividends is not given and therefore we shall first ascertain it.

Illustration

The dividend yield of a company is 12% and the expected dividend is Sh. 10. Currently the company

pays a dividend of Sh. 9.5.

Calculate Ke

Solution

Dividend Yield = 𝐷𝑃𝑆

𝑀𝑃𝑆

0.12 = 9.5

𝑥

0.12x = 9.5

x = 9.5

0.12

x = 79.17

x = market price per share = P0

Determine growth rate as follows:

D1 = Do (1 + g)

10 = 9.5 (1 + g)

10 = 9.5 + 9.5g

10 – 9.5 = 9.5g

0.5 = 9.5g therefore g = 0.5

9.5 = 0.05

g = 0.05 × 100 = 5%

Note: D1 = Expected dividend = 10

Ke = 𝐷1

𝑃𝑜+ 𝑔=

10

79.17 + 0.05 = 0.1763 x 100 = 17.63%

Cost of Preference Shares (Kp)

It is the minimum rate of return that should be obtained from the projects that are fully financed by

preference share capital so as to generate sufficient cash flows required to pay back to preference

shareholders.

It is therefore the minimum rate of return required by preference shareholders.

Kp is the discount factor that equate the present value of expected dividends to the current market

value per preference share i.e.

Page 116

Value of preference share (Vp) = Preference dividends x PVIFAr%,∝

Recall: 𝑃𝑉𝐼𝐹𝐴𝑟%∝ = 1

𝑟

𝑃𝑉𝐼𝐹𝐴𝑘𝑝∝ = 1

𝐾𝑝

Vp = Preference dividends x 1

𝐾𝑝

Vp x Kp = Preference dividends

Kp = 𝑃𝑟𝑒𝑓 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑

𝑉𝑝

Where:

Kp – Cost of preference shares

Vp – Current market price per share

NB: In case the shares are issued at a discount or at a floatation/issue cost, such costs would reduce

the market value hence

Kp = 𝑃𝑟𝑒𝑓 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠

𝑉𝑝−𝑑−𝑓

Where d = discount on issue

f = floatation issue costs

Cost of Long Term Debt

(Kd)/Debentures

It is the minimum rate of return required by debenture holders /providers of long term debt finance.

It is basically the discount factor that equates the present value of expected interest and the

redemption value if any to the current market value of debentures.

Cost of Irredeemable Debentures

This kind of debentures are rare and they attract a proceed amount of interest to infinity i.e. they do

not mature.

The cost of irredeemable debentures before tax cost can be simply calculated using the formula;

Kd = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡

𝐵𝑜

Where

Kd – Before tax costs or debt

Bo – Current market value/Intrinsic value per debenture.

NB:

The ideal cost of debt is the after tax cost.

Page 117

This is because the interest finance cost is an allowable expense for tax purposes. The before tax cost

should therefore be adjusted for tax purposes i.e.

Kd = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡

𝐵𝑜(1 − 𝑇)

Where T = Tax Rate

Cost of Redeemable Debentures

These debentures attract annual interest for a given period after which they are redeemed upon

maturity.

The before tax cost of redeemable debentures can be simply calculated using the formula of the Bonds

Yield to Maturity (YTM)

YTM = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡+ 1 𝑛⁄ [𝐹−𝐵𝑜]

1

2[𝐹+𝐵𝑜]

N = No of years to maturity

F = Face/Par/Nominal Value

Bo = Current Market Value/Redemption Value

Kd = YTM [1-T]

Kd = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡+ 1 𝑛⁄ [𝐹−𝐵𝑜]

1

2[𝐹+𝐵𝑜]

[1 − 𝑇]

Weighted Average Cost of Capital

A firm’s Weighted Average Cost of Capital (WACC) represents its blended cost of capital across all

sources, including common shares, preferred shares, and debt. The cost of each type of capital is

weighted by its percentage of total capital and they are added together.

It is a combined cost made up by component costs that are utilised to finance the existing projects.

The contribution of the component costs should always be made in a manner that is meant to reduce

the overall cost to the company at large.

It is worked out using market value weight and not the book value weights.

Retained earnings is excluded from the computation since it represents reserve of the company which

will be reinvested and reflected in the market value.

Here is the basic formula for weighted average cost of capital:

WACC = ((E/V) * Ke) + [((D/V) * Kd)*(1-T)]

E = Market value of the company's equity

D = Market value of the company's debt

V = Total Market Value of the company (E + D)

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Ke = Cost of Equity

Kd = Cost of Debt

T= Tax Rate

Limitation of WACC

i) The use of WACC as a discounting rate assumes that the company’s mix of longterm sources of

finance does not change.

ii) The funds required to finance a given project does not come from all the sources of finance but

from a specific component cost which should be used in evaluating the project and not the

WACC.

iii) WACC assumes that the projects to be undertaken by the company in future will be of the same

risk as the company’s current projects. However, the risk level of a company keeps on changing.

iv) WACC assumes that the company’s dividend pay out ratio will remain constant. However, in

practice the company’s dividend policy keeps on changing from one period to another.

Illustration

The capital structure of Ukulima Ltd is as shown below.

Sh. 000

Common Equity (Sh. 25 par)

8% preference shares (Sh. 25 par)

10% Debentures (Sh. 100 par)

5000

4000

4000

13000

The Current Market Value per Ordinary Share, Preference Share and Long Term Debt is Sh. 15, Sh.

29 and Sh. 110 respectively.

The company has just paid a dividend of Sh. 12 with a growth rate of 8% p.a. Issue of preference

shares attract a discount of Sh. 5 and debentures attract a discount of Sh. 10 each.

Tax rate is at 30%.

Required:

Calculate company’s Weighted Average Cost of Capital (WACC).

Solution

Steps of calculating WACC

(i) Calculate the component costs i.e. Ke, Kp and Kd

(ii) Calculate the weight proportion of each capital component in the capital structure.

NB: In calculating the weight, we can either use book values (balance sheet values), market values or

protected values.

Normally, market values are more ideal and should therefore be used in case the question is silent.

(iii) Multiply the component cost by its weight to get the weighted component cost.

(iv) Add together the weighted component costs to arrive at WACC i.e. WACC = Weke + wpkp +

Wdkd(1 – T)

Component Costs

Page 119

Cost of equity (ke)

Ke = 𝐷1

𝑃𝑜+ 𝑔

NB: D1 = D0 (1 + g) = 12(1.08)

D0 = historical dividend = 12.

𝟏𝟐(𝟏.𝟎𝟖)

𝟏𝟓+ 0.08 = 0.944 × 100 = 94.4%

Cost of preference shares (kp)

Kp = 𝑃𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠

𝑉𝑝−𝑑 =

2

29−5= 0.0833 x 100 = 8.33%

Preference dividend = 8% x 25 = 2

Cost of debt (kd)

Kd = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡

𝐵𝑜(1 − 𝑇) =

10

110−10 (1 − 0.3) = 0.07 x 100 = 7%

Interest = 10% of par = 10% x 100 = 10

B0 = market value of Bond = 110

Market Value

Source Amount

Sh. 000

Weight Cost Weight × Cost

Equity

8% Preference

Shares

10% Debentures

3000 (w1)

4640 (w2)

4400 (w3)

12040

0.2492

0.3854

0.3654

0.944

0.0833

0.07

0.2352

0.0321

0.0256

0.2929 × 100

= 29.29% ≃ 30%

Alternatively

WACC = WeKe + WpKp + WdKd [1-T]

3,000

12,040 × 94.4% +

4640

12040 × 8.33% +

4400

12040 × 7%

= 0.2929 × 100 = 29.29%

Workings = Total market values

W1 = number of ordinary shares x market price per share

= 5,000

25 x 15 = 3,000

W2 = 4,000

25 x 29 = 4,940

W3 = 4,000

100 x 110 = 4,400

Illustration

Assuming the above debentures were redeemable in 15 years, what would be their cost.

Page 120

Solution

First compute the yield to maturity

Yield to maturity (YTM) = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡+

1

𝑛 (𝐹−𝐵𝑜)

1

2 (𝐹+𝐵𝑜)

Formular for redeemable debentures

Where

F = face value

B0 = market value of bond less costs to sell = 110 – 10 = 100

T = Tax

10+ 0

1

2 (100+100)

10+ −01

2 𝑥 200

= 10%

After tax cost (kd (1 + T) = 10% (1 – 0.3) = 7%

NB: When the debentures or preference shares are selling at par, their before tax cost would be the

coupon rate.

Marginal Cost of Capital (MCC)

It is the cost of incremental new funds that are used to undertake the firm’s new projects i.e. it is a

futuristic cost.

WACC is the cost incurred by the company to raise funds that are used to implement existing projects

i.e. it is a historical cost.

In calculating WMCC, marginal book value weights are used.

The marginal weights are the proportions of capital components calculated from the optimal capital

structure.

Retained earnings is included in the computation because it must be utilised up to a point of

exhaustion before raising additional finance through issue of ordinary share.

Breakpoint

It is the level of total financing (100%) at which amount available from a cheaper source in a given

capital component are fully utilised.

Breakpoint arises when a given capital component can generate more than one source of capital e.g.

from equity we can get retained earnings and ordinary shares.

A breakpoint will occur if there is an increase in the cost of capital.

Breakpoint for retained earnings = 𝑟𝑒𝑡𝑎𝑖𝑛𝑒𝑑 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠

optimal equity proportion

Page 121

Breakpoint for Debt capital= 𝐶ℎ𝑒𝑎𝑝𝑒𝑟 𝑑𝑒𝑏𝑡𝑠

optimal debt proportion

Illustration

Mapema Ltd has the following capital structure which it considers optimal:

Source of capital Amount

Sh. “million”

Ordinary share capital

Preference share capital

Long term debt

Total

90.0

22.5

37.5

150.0

Mapema Ltd expects an after tax income of sh.5,143,000 in the next financial year. The company has

a policy of 30% of its earnings as dividends. Investors expect dividends to grow at an annual rate of

9% indefinitely. The dividends paid by the company were sh.5.40 per share. The company’s ordinary

shares currently sell on the stock market at sh.90 per share. The company can obtain additional

financing in the financial markets as follows:

Long-term debt

Up to sh.7.5 million of long-term debt can be obtained at an interest rate of 12%; long-term debt in the

range of sh.7.5 million to sh.15 million must carry an interest rate of 14%; and all long-term debt over

sh.15 million will have an interest rate of 16%. The corporate tax rate is 30% and interest on long-

term debt is tax allowable.

Ordinary shares

New ordinary shares of up to sh.18 million can be raised at sh.81 per share. To issue additional shares

above sh.18 million floatation cost of sh.18 per share must be incurred.

Preference shares

New preference shares with a par value of sh.100 can be issued and the dividend rate is 11%.

However, a floatation cost of 5% of the par value per share must be incurred for all preference shares

up to sh.11.25 million. Additional preference shares (above sh.11.25 million) can be raised at a

floatation cost of sh.10 per share.

The investment opportunities available to the company are as shown below:

Investment Outlay

Sh.

Annual net cash flow

Sh.

Life (years) Internal rate of

return (IRR)

(%)

I

II

III

IV

V

15,000,000

15,000,000

15,000,000

30,000,000

30,000,000

3,286,800

4,731,630

3,255,270

5,684,220

8,141,760

7

5

8

10

6

12.0

17.4

14.2

16.0

?

Required:

(a) Determine the break points in the marginal cost of capital (MCC) schedule.

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(b) Calculate the weighted average cost of capital (WACC) in the intervals between the break points

in the MCC schedule.

(c) Calculate the internal rate of return (IRR) for project V.

(d) Construct an investment opportunity curve (IOC) marginal cost of capital (MCC) schedule and

indicate which project(s) should be accepted or rejected.

Solution

(a) Weights (obtained from the capital structure)

Weight of Equity = 90

150 x 100 = 60%

Weight of Preference share capital = 22.5

150 x 100 = 15%

Weight of Debt =37.5

150 x 100 = 25%

Equity

Retained earnings = 70% x 5143,000 = 3,600,100

Breakpoint of retained earnings = 𝐴𝑚𝑜𝑢𝑛𝑡

𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦

= 3600100

0.6= 6,000,167 ksh

Breakpoint (ordinary share upto 18,000,000) = 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑎𝑚𝑜𝑢𝑛𝑡

𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑑𝑒𝑏𝑡

= 18000000+3600100

0.6

= 36,000167 kshs

Breakpoint (for ordinary shares beyond 18000000). The upper limit is not provided

There if no limit and therefore there is no breakpoint.

Breakpoint of debts (weight of debt = 25%

Breakpoint (upto 7,500,000) = 𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑎𝑚𝑜𝑢𝑛𝑡

𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦

= 7,500,000

25% = 30m

Breakpoint (7.5m up to 15m) = 15,000,000

25% = 60m

Breakpoint (Above 15m) = No limit and therefore no break point.

Breakpoint of preference (weight of preference = 15%)

Breakpoint (up to 11.25m) = 11,250,000

15% = 75,000,000 Ksh

Breakpoint (preference above 11.25m) = No limit, no break point.

(b) WACC

Cost of Equity = 𝐷𝑜 (1+𝑔)

𝑃𝑜−𝑑+ 𝑔

1) Cost of (ke1)

Retained Earnings = 5.40 (1.09)

90+ 0.09

0.0654 + 0.09 = 0.1554 x 100

= 15.54%

2) Cost of (ke2) up to ksh.18,000,000

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Ordinary Shares = 5.40 (1.09)

81+ 0.09

0.0727 + 0.09 = 0.1627 × 100 = 16.27%

3) Cost of ordinary shares above 18,000,000 (ke3)

= 5.4(1.09)

(90−18) + 0.09= 0.17175 = 17.2%

Cost of Long term debt = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡

𝐵𝑜[1 − 𝑇] = cost before tax (I – T)

12% long term debt (kd1) = 0.12 (1 – 0.3)

= 0.084 x 100 = 8.4%

14% long term debt (kd2)= 0.14 (0.7)

= 0.098 x 100 = 9.8%

16% long term debt (kd3) = 0.16 (0.7)

= 0.112 x 100 = 11.2%

Preferences Shares = 𝑃𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠

𝑉𝑝−𝑑𝑓

Upto 11.25m (Floatation cost = 5% x 100 = 5)

Kp1 = 11

100−5 = 0.1158 x 100 = 11.58%

Over 11.25m (Floatation cost = 10)

Kp2 = 11

100−10 = 0.1222 x 100 = 12.22%

Summary

Equity

Bp1 = 6,000,167 ke1 = 15.54%

Bp2 = 36,000,167 ke2 = 16.27%

Ke3 = 17.2%

Debt

Bp1 = 30,000,000

Bp2 = 60m

Kd1 = 8.4%

Kd2 = 9.8%

Kd3 = 11.2%

Preference

Bp1 = 75,000,000

Kp1 = 11.58%

Kp2 = 12.22%

Preference dividend = 11% of par value

11% × 100 = 11

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Financing Range schedule

Breakpoint

(Ascending

order)

Range We = 0.6

Ke

Wp = 0.15

Kp

Wd = 0.25

Kd

WMCC

6,000,167 O up to 6000167 15.54% 11.58% 8.4% W1 = 13.16

30,000,000 6000167 up to 30,000,000 16.27% 11.58% 8.4% 13.60

36,000,167 30,000,000 up to 36,000,167 16.27% 11.58% 9.8% 13.95

60,000,000 36,000,167 up to 60,000,000 17.2% 11.58% 9.8% 14.51%

75,000,000 60,000,000 up to 75,000,000 17.2% 11.58% 11.2% 14.86

Over 75,000,000 17.2% 12.22% 11.2% 14.95

(c) IRR of project V

8,141,760 × PVIFAr%6yrs – 30,000,000 = 0

8141760 𝑃𝑉𝐼𝐹𝐴𝑟%6𝑦𝑟𝑠

8141760 =

30,000,000

8141760

PVIFAr%6yrs = 3.6847 = 16%

(d) Rank the project first

IOC/MCC Schedule

Ranks IRR Cost Cumulative

cost

MCC Decision

II

IV 1

2

17.4%

16%

15,000,000

30,000,000

15,000,000

45,000,000

13.6%

14.5%

Accepted

Accepted

18

17

16

15

14

13

12

15 45 75 90 105

II IV V III I

Optimal loan and

accept II, IV & V

Cumulative

costs

IRR 12%

13.6%

14.51

16% 16%

14.9% 14.9%

14.2%

17.4%

14.86 MCC

Page 125

V

III

I

3

4

5

16%

14.2%

12%

30,000,000

15,000,000

15,000,000

75,000,000

90,000,000

105,000,000

14.86%

14.95%

14.95%

Accepted

Rejected

Rejected

The projects whose IRR is more than MCC are accepted hence projects II, IV and V should be

implemented. The optimal amount to be borrowed to finance the 3 projects is Ksh.7,000,000.

Capital Structure Financing Decisions

Capital structure is the mix of debt, preference share capital and equity that is used by a firm to

finance its long term investment activities.

The capital structure decisions are always aimed at maximising the value of the firm and minimising

the overall cost of capital (WACC).

Normally, the optimal capital structure should be planned for by each company. This mix of debt and

equity minimises the cost of capital (WACC) and maximises the shareholders’ wealth/value of the

firm.

OPTIMUM CAPITAL STRUCTURE

Optimum capital structure is the capital structure at which the weighted average cost of capital is

minimum and thereby the value of the firm is maximum. Optimum capital structure may be defined as

the capital structure or combination of debt and equity that leads to the maximum value of the firm.

Objectives of Capital Structure Decision of capital structure aims at the following two important

objectives:

1. Maximize the value of the firm.

2. Minimize the overall cost of capital

Factors that influence capital structure decisions/financing decisions.

Component cost

Some firms could avoid equity due to its high cost as compared to debt.

Debt is always expected to be less expensive since it is more secure from the investments point of

view hence they would demand a lower return compared to equity. Moreover, the company saves

some tax due to interest associated with debt.

Availability of Assets to Pledge as Security

Companies that have disposable assets that can be pledged as collateral for a loan can increase their

capital structure by raising more debt unlike a company without such a facility.

Tax rate

Page 126

Firms that operate in economics of higher taxes can have tax advantage of the system and use more of

debt in their capital structure to save some taxes from the interest known as interest tax shield i.e. the

finance cost is a tax-deductible expense.

Manager’s attitude to risk

Risk takers would always aim to expand which will be financed by increasing the capital structure. At

times, the management that are risk takers may even opt to raise more funds through debt due to fewer

procedures involved.

Risk averse management may not increase their capital structure since they would be too reluctant to

expand more so increase their debt due to the fear of financial gearing risk.

Growth stage of the Firm.

Any company has to raise more funds so as to support/finance its expansion/growth in terms of

revenue and total assets.

Flexibility

A capital structure that can be easily altered/changed with a minimum cost and delays shall be

adopted by many companies.

CAPITAL STRUCTURE THEORIES Some commentators believe that an optimal mix of finance exists at which the company’s cost of

capital will be minimized.

When we consider the capital structure decision, the question arises of whether there is an optimal

mix of equity and debt which a company should try to achieve. Under the traditional view there is an

optimal capital mix at which the average cost of capital, weighted according to the different forms of

capital employed, is minimized.

However, the alternative view of Modigliani and Miller is that the firm’s overall weighted average

cost of capital is not influenced by changes in its capital structure.

Different theories were formulated in different years with the aim of understanding the effect of debt.

Financing in the company’s capital structure.

They include:

i) Traditional Theories

ii) Net Income Theory (NI)

iii) Net Operating Income Approach Theory (NOI)

iv) Modigliani and Miller (MM) Prepositions

TRADITIONAL THEORY

Under the traditional theory of cost of capital, the cost declines initially and then rises as gearing

increases. The optimal capital structure will be the point at which WACC is lowest.

The traditional view of structure is that there is an optimal capital structure and the company can

increase its total value by a suitable use of debt finance in its capital structure.

The assumptions on which this theory is based are as follows:

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(a) The company pays out all its earnings as dividends.

(b) The gearing of the company can be changed immediately by issuing debt to repurchase

shares, or by issuing shares to repurchase debt. There are not transaction costs for issues.

(c) The earnings of the company are expected to remain constant in perpetuity and all

investors share the same expectations about these future earnings.

(d) Business risk is also constant, regardless of how the company invests its funds.

(e) Taxation, for the time being, is ignored.

The traditional view is as follows:

(a) As the level of gearing increases, the cost of debt remains unchanged up to a certain level

of gearing. Beyond this level, the cost of debt will increase.

(b) The cost of equity rises as the level of gearing increases and financial risk increases.

There is a non-linear relationship between the cost of equity and gearing.

(c) The weighted average cost of capital does not remain constant, but rather falls initially as

the proportion of debt capital increases, and then begins to increase as the rising cost of

equity (and possibly of debt) becomes more significant.

(d) The optimum level of gearing is where the company’s weighted average cost of capital is

minimized.

The traditional view about the cost of capital is illustrated in the following figure. It shows that the

weighted average cost of capital will be minimized at a particular level of gearing P.

Ke

K0

Kd

Level of gearing 0

Cost of

capital

P

Page 128

Where

Ke is the cost of equity in the geared company

Kd is the cost of debt

K0 is the weighted average cost of capital

The traditional view is that the weighted average cost of capital, when plotted against the level of

gearing is saucer shaped. The optimum capital structure is where the weighted average cost of capital

is lowest, at point P.

Conclusion

Traditional theory therefore, states that there exists a capital structure which maximise the value of a

firm and minimises the cost of capital (WACC) at an optimal debt level.

PECKING ORDER THEORY

Pecking order theory has been developed as an alternative to traditional theory. It states that firms will

prefer retained earnings to any other source of finance, and then will choose debt, and last of all

equity.

The order of preference will be:

Retained earnings

Straight debt

Convertible debt

Preference shares

Equity shares

Reasons for following pecking order

(a) It is easier to use retained earnings than go to the trouble of obtaining external finance and have to

live up to the demands of external finance providers.

(b) There are no issue costs if retained earnings are used, and the issue costs of debt are lower than

those of equity.

(c) Investors prefer a safer security, which is debt with its guaranteed income and priority on

liquidation.

(d) Some managers believe that debt issues have a better signalling effect than equity because the

market believes that managers are better informed about shares’ true worth than the market itself

is. Their view is the market will interpret debt issues as a sign of confidence, that businesses are

confident of making sufficient profits to fulfil their obligations on debt and that they believe that

the shares are undervalued.

By contrast the market will interpret equity issues as a measure of last resort that managers believe

that equity is currently overvalued and hence are trying to achieve high proceeds whilst they can.

However an issue of debt may imply a similar lack of confidence to an issue of equity; managers may

issue debt when they believe that the cost of debt is low due to the market underestimating the risk of

default and hence undervaluing, the risk premium in the cost of debt. If the market recognizes this

lack of confidence, it is likely to respond by raising the cost of debt.

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The main consequence in this situation will be to reinforce a preference for using retained earnings

first. However debt (particularly less risky, secured debt) will be the next source as the market feels

more confident about valuing it than more risky debt or equity.

Behavioural theories

This theory states that the company shall always aim at maintaining a capital structure that is closely

related to that of an average firm in the industry.

Benchmark Theory

This theory suggests that firms will identify a trader or another company in the market and adopt a

similar capital structure.

Post Experience Concept

It is an important influence in the capital structure which is based on accumulative knowledge and

experience about the past that the managers will select an optimal capital structure that will give them

less trouble i.e. higher returns and lower costs.

Traditional and Static Trade-Off Theory

Static trade off view develops the traditional theory and states that firms will always aim from an ideal

capital structure and will issue debt and equity depending on their current needs and preferences e.g.

A company can issue redeemable debt so as to take advantage of the tax savings on interest and also

to invest in a profitable project after which it would redeem the debt so as to move back to its optimal

capital structure in the long run.

Practical implications of the static-trade-off theory

1. Firms with low distress costs should load up on debt to get the tax shield (these are firms with

mostly tangible assets; Example: airlines, real estate holding companies).

2. Firms with high distress costs (firms with mostly intangible assets) should follow more

conservative debt financing policies; Example: high-tech companies).

3. Firms with a high probability of financial distress should go for capital structures that minimize

the costs of financial distress:

- Avoid too much debt

- If need debt, go for an easy-to-reorganize debt structure: – Banks rather than many

bondholders – Few rather than many banks – Few rather than many classes of debt.

Problems with the static trade-off theory

1. Ignores potential gains from market imperfections

Opportunities to issue securities on favorable terms (e.g., government guarantees)

Demand for certain securities that are not available in the market.

2. Ignores information problems

3. Ignores incentive effects of leverage — LBO example

4. Firms do not seem to have well-defined debt ratios.

Page 130

Generally, the capital structure theories have the following assumptions:

1. There are no corporate taxes.

2. The firms use only 2 sources of financing namely perpetual debts and equity shares

3. The firms pay 100% of the earnings as dividend. This means that the dividend pay-out ratio is

100% and there are no earnings that are retained by the firms.

4. The total assets are given which do not change and the investment decisions are assumed to be

constant.

5. Business risk is constant over time and it is assumed that it is independent of the capital structure.

6. The firm has a perpetual life.

7. The firms earnings before interest and taxes are not expected to grow.

8. The firms total financing remains constant. The firms degree of leverage can be altered either by

selling shares and to retire the debt using the proceeds or by raising more debt and reduce the

equity financing.

9. All the investors are assumed to have the same subjective probability distribution of the future

expected operating profits for a given firm.

NET INCOME APPROACH (NI)

Net Income theory was introduced by David Durand. According to this approach, the capital

structure decision is relevant to the valuation of the firm. This means that a change in the financial

leverage will automatically lead to a corresponding change in the overall cost of capital as well as the

total value of the firm. According to Net income approach, if the financial leverage (debt) increases,

the weighted average cost of capital decreases and the value of the firm and the market price of the

equity shares increases. Similarly, if the financial leverage decreases, the weighted average cost of

capital increases and the value of the firm and the market price of the equity shares decreases.

Net income approach also:

Portrays the influence of leverage/gearing on the value of the firm i.e. how debt affects the value of

the firm.

Leverage is associated with the use of debt in the capital structure.

Cost of debt is always identified to be cheaper than equity which means that debt would reduce the

overall (WACC) cost of capital and increase the total value of the firm.

Net income is based on the following assumptions: -

i) The company is financed by only 2 types of capital i.e. debt and equity.

ii) The company operates in an environment of no taxes

iii) All earnings are paid out as dividends i.e. Dividend Pay Out Ratio = 100% (EPS = DPS)

iv) Debt is always cheaper than equity

v) The use of debt in the capital structure does not change the risk attitude of the investor.

Recall: (For a non-growth firm)

Ke = 𝐷1

𝑃𝑜

Ke = 𝐸𝑃𝑆

𝑃𝑜[𝑃𝑎𝑦𝑜𝑢𝑡 𝑟𝑎𝑡𝑖𝑜 = 100%] When payout ratio = 100%

Then dividends per share = EPS

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Ke = 𝐸𝐵𝐼𝑇−𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡