2 Robert Alan Hill

The World of Modigliani and Miller

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3 The World of Modigliani and Miller

1st edition© 2015 Robert Alan Hill &

bookboon.com

ISBN 978-87-403-1062-7Download free eBooks at bookboon.com

4 Contents

Contents

Part One: An Introduction

91

An Overview

101.1 ˜e Foundations of Finance: An Overview

111.2 ˜e Development of Financial Analysis

121.3 Questions to Consider

171.4 Fisher™s Legacy and Modigliani-Miller

181.5 Summary and Conclusions

221.6 Selected References

22 Part Two: ˜e Dividend Decision

242

How to Value a Share

252.1 ˜e Capitalisation Concept

272.2 ˜e Capitalisation of Dividends and Earnings

292.3 ˜e Capitalisation of Current Maintainable Yield

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5 Contents

2.4 ˜e Capitalisation of Earnings

322.5 Summary and Conclusions

352.6 Selected References

353 ˜e Role of Dividend Policy

363.1 ˜e Gordon Growth Model

363.2 Gordon™s ‚Bird in the Hand™ Model

393.3 Summary and Conclusions

423.4 Selected References

424

MM and Dividends

434.1 ˜e MM Dividend Hypothesis

434.2 ˜e MM Hypothesis and Shareholder Reaction

454.3 ˜e MM Hypothesis: A Corporate Perspective

474.4 Summary and Conclusions

514.5 Selected References

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6 Contents

Part ˜ree: ˜e Finance Decision

525

Debt Valuation and the Cost of Capital

535.1 Capital Costs and Gearing (Leverage): An Overview

545.2 ˜e Value of Debt Capital and Capital Cost

565.3 ˜e Tax-Deductibility of Debt

595.4 ˜e Impact of Issue Costs on Equity and Debt

625.5 Summary and Conclusions

655.6 Selected References

666

Capital Gearing and the Cost of Capital

676.1 ˜e Weighted Average Cost of Capital (WACC)

676.2 WACC Assumptions

696.3 ˜e Real-World Problems of WACC Estimation

716.4 Summary and Conclusions

766.5 Selected Reference

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7 Contents

7 MM and Capital Structure

797.1

Capital Structure, Equity Return and Leverage

807.2 Capital Structure and the Law of One Price

857.3 MM and Proposition I (the Arbitrage Process)

927.4 MM and Real World Considerations

957.5 Summary and Conclusions

997.6 Selected References

101 Part Four: ˜e Portfolio Decision

1028 Portfolio Selection and Risk

1038.1 Modern Portfolio ˜eory and Markowitz

1088.2 Modern Portfolio ˜eory and the Beta Factor

1098.3 Modern Portfolio ˜eory and the CAPM

1118.4 Summary and Conclusions

1158.5 Selected References

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8 Contents

9 MM and the CAPM

1199.1 Capital Budgeting and the CAPM

1199.2 ˜e Estimation of Project Betas

1219.3 Capital Gearing and the Beta Factor

1269.4 Capital Gearing and the CAPM

1309.5 Modigliani-Miller and the CAPM

1329.6 Summary and Conclusions

1389.7 Selected References

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9 Part One:

An Introduction

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10 An Overview

1

An Overview

Introduction

Financial analysis has never been an exact science. Occasionally, the theoretical models upon which

it is based are even ﬁbadﬂ science. ˜e root cause is that economic decisions undertaken in a

real

world of uncertainty are invariably characterised by

hypothetical

human behaviour, for which there is

little

empirical

evidence. ˜us, a ˚nancial model may satisfy a fundamental requirement of all theory

construction. It is based on

logical

reasoning. But if the

objectives

are too divorced from

reality

, or

underpinned by

simplifying assumptions

that rationalise

complex

phenomena

, the

analytical conclusions

may be invalid.

Nevertheless, all theories, whether bad or good, still serve a useful role.

Ł At worst, they provide a benchmark for future development to overcome their de˚ciencies,

which may require correction, or even a thorough revision of objectives.

Ł At best, they serve to remind us that the ultimate question is not whether a theory is an

abstraction of the real world. But does it work?

˜e purpose of this study is to illustrate the development of basic ˚nancial theory and what it o˛ers,

with speci˚c reference to the seminal work of two Nobel Prize economists who came to prominence in

the 1950s and have dominated the world of ˚nance ever since:

Franco Modigliani (1918Œ2003)

Merton H. Miller (1923Œ2000)

˜e text™s inspiration is based on readership feedback from my

bookboon

series, which welcomed

various explanations of Modigliani and Miller™s controversial hypothesis that

identical

˚nancial assets

(for example, two companies, their individual shares, or capital projects) cannot be valued and traded

at

di˜erent

prices.

Many readers also mentioned that this application of the economic ﬁlaw of one priceﬂ, which permeates

the series, concerning the

irrelevance

of dividend policy, capital structure and its portfolio theory

implications, should be published in a single volume to focus their studies.

I agree, whole-heartedly.

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11 An Overview

All too o˝en, throughout my academic career, I have observed that Modigliani and Miller™s body of work

is a ﬁwall of worryﬂ that ˚nance students must climb when revising for examinations. Consequently, it is

frequently regarded as a topic best avoided (even though it crops up in di˛erent questions) and is soon

forgotten when they enter the real world of work.

If you don™t want to fall into this trap, let us therefore return to ˚rst principles and remind ourselves

of some signi˚cant developments in modern ˚nance theory, which predate Modigliani and Miller,

concerning its objectives, assumptions and conclusions.

Having set the scene, we can then evaluate the positive theoretical contribution of Modigliani and Miller

(MM henceforth) to the academic debate and what it o˛ers as a springboard for sound ˚nancial analysis.

As we shall discover, no one should doubt that MM™s original conclusions are logically conceived, given

their rigorous theoretical assumptions. ˜e question we can then address in this text™s subsequent Exercise

companion is the extent to which MM™s theoretical conclusions still apply, once their basic assumptions

are relaxed to introduce greater realism and subsequent empirical research.

1.1 The Foundations of Finance: An Overview

Today, most theorists still begin their analyses of corporate investment and ˚nancial behaviour with the

following over-arching

normative

objective.

The maximisation of shareholders™ wealth, using ordinary share price (common stock) as a universal metric, based on a

managerial interpretation of their ﬁrationalﬂ and ﬁrisk-averseﬂ expectations (by which we mean the receipt of more money

rather than less, and more money earlier).

Management model shareholder expectations using the ﬁtime value of moneyﬂ concept (the value of

money over time, irrespective of in˙ation) determined by borrowing-lending rates. Using net present

value (NPV) maximisation techniques, their strategy is to invest in a portfolio of capital projects that

delivers the ﬁhighest

absolute

pro˚t at minimum riskﬂ.

˜is model has a long-standing academic pedigree.

It begins with the ﬁSeparation ˜eoremﬂ of Irving Fisher (1930) that assumes

perfect

capital markets,

characterised by perfect knowledge, freedom of information and ﬁno barriers to tradeﬂ (for example,

innumerable investors, uniform borrowing-lending rates, tax neutrality and zero transaction costs).

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12 An Overview

Subject to the constraint that management™s discount rate for project appraisal at least equals the

shareholders™

opportunity

cost of capital (or desired return) to be earned elsewhere on comparable

investments of equivalent risk:

Ł ˜e wealth and consumption (dividend) preferences of all shareholders are satis˚ed by the

managerial investment and ˚nancing policies of the company that they own.

By de˚nition, because

perfect

markets are also

e˚cient

, whereby market participants (including

management) respond

instantaneously

to events as they unfold, it follows, that:

Ł Shares should always be correctly priced at their

intrinsic

true value.

Ł All shareholders earn a return commensurate with the risk of their investment and so wealth

is maximised.

Decades later, Fisher™s analysis and speci˚cally the importance of his investment constraint, were

formalised by the ﬁAgency ˜eoryﬂ of Jenson and Meckling (1976). ˜ey explained that even though

corporate (shareholder)

ownership

is divorced from managerial

control

:The

agent (management) motivated by self-preservation should always act in the best interests of the

principal (shareholder). Otherwise, any failure to satisfy shareholder expectations may result in their replacement.

˜e Eˆcient Market Hypothesis (EMH) of the Nobel Prize winning Laureate Eugene Fama (1965) also

lent further credence to Fisher™s Separation ˜eorem. As he observed, history tells us that capital markets

or not ﬁperfectﬂ. For example, access to information may incur costs and there are barriers to trade. But

if we assume that they are ﬁreasonably eˆcientﬂ:

The consequence of decisions undertaken by management on behalf of their shareholders (the

agency

principle) will

eventually be communicated to market participants. So, share price adjusts quickly but not instantaneously to a new

equilibrium value in response to ﬁtechnicalﬂ and ﬁfundamentalﬂ analyses of historical data, current events and trending

media news.

1.2 The Development of Financial Analysis

As a convenient benchmark for subsequent analyses and critiques of modern ˚nance theory, all the texts

in my

bookboon

series begin with this

idealised

picture of market behaviour.

˜e

majority

of investors are rational and risk-averse, motivated by

self-interest

, operating in

reasonably

eˆcient capital markets characterised by a

relatively

free ˙ow of information and

surmountable

barriers

to trade.

If we also assume a world of certainty, where future events can be speci˚ed in advance, it follows that

investors can formally analyse one course of action in relation to another for the purpose of wealth

maximisation with con˚dence.

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13 An Overview

For an

all-equity

˚rm ˚nanced by ordinary shares (common stock) summarised in Figure 1.1 below,

where the ownership of corporate assets is divorced from control (the

agency

principle), we can formally

de˚ne and model the

normative

goal of strategic ˚nancial management under conditions of certainty as:

Ł ˜e implementation of optimum investment and ˚nancing decisions using net present value

(NPV) maximisation techniques to generate the highest money pro˚ts from all a ˚rm™s projects

in the form of retentions and distributions. ˜ese should satisfy the ˚rm™s

existing

owners (a

multiplicity of shareholders) and

prospective

equity investors who de˚ne the capital market,

thereby maximising share price.

Figure 1.1:

The Mixed Market Economy

Over their life, individual projects should eventually generate net cash ˙ows that

exceed

their overall

cost of funds to create wealth. ˜is future

positive

net terminal value (NTV) is equivalent to a

positive

NPV, expressed in today™s terms, de˚ned by the project discount rate using the time value of money.

Even when modern ˚nancial theory moves from a risk-free world to one of uncertainty, where

more

than one future outcome

is possible

, this analysis remains the bedrock of rational investment behaviour.

Providing markets are reasonably eˆcient, all news (good or bad) is soon absorbed by the market,

such that:

Ł Short-term

, you win some, you lose some.

Ł Long-term

, the market provides returns commensurate with their risk.

Ł Overall

, you cannot ﬁbeatﬂ the market.

Without permanent access to ﬁinsider informationﬂ (which is illegal) investment strategies using ﬁpublicﬂ

information, such as share price listings, corporate and analyst reports, plus press and media comment,

represent a ﬁfairﬂ game for all (

i.e. a martingale

). Download free eBooks at bookboon.com

14 An Overview

As I have also illustrated throughout my

bookboon

series with reference to volatile, historical events:

from Dutch ﬁtulip maniaﬂ (1637) to the 1929 and 1987 stock market crashes, the millennium dot.com

bubble, global ˚nancial meltdown (2008), subsequent Euro crises and the 2015 Dow Jones and FTSE

100 (Footsie) record highs:

Even the most sophisticated ˜nancial institutions and private investors, with the time, money, ˜nancial and ˜scal

expertise to analyse

all public information, have failed spectacularly to identify trends.

So, the only way foreword for uncertain investors is to accept that knowledge of the past (or even current

events) is no guide to future plans. It is already incorporated into the latest share price listings. And this

is where Fama™s EMH (

op.cit

.) provides a lifeline.

Taking his

linear

view of society, where ﬁeˆcient markets have no memoryﬂ and participants lack perfect

foresight, it is still possible to de˚ne

expected

investor returns for a given level of risk, using the techniques

of ﬁclassicalﬂ statistical analysis (

Quants

). Assuming a ˚rm™s project or stock market returns are linear, they are

random

variables

that conform to a

ﬁnormalﬂ distribution. For every level of risk, there is an investment outcome with the highest expected

return. For every expected return there is an investment outcome with the lowest expected risk. Using

mean-variance analysis, the standard deviation calibrates these risk-return pro˚les and the likelihood

of them occurring, based on probability analysis and con˚dence limits. Wealth maximisation equals the

maximisation of investor

utility

using this trade-o˛, plotted as an

indi˜erence

curve, which calibrates

the

certainty equivalence

associated with the maximisation of an investment™s

expected

NPV (ENPV).

According to Modern Portfolio ˜eory (MPT) and the pioneering work of Markowitz (1952), Tobin

(1958) and Sharpe (1963), if numerous investments are then combined into an optimum portfolio,

management (or any investor) can also plot an ﬁeˆciency frontierﬂ using Quants and evaluate a new

investment™s inclusion into the mix, according to their risk-return pro˚le (utility curve) relative to their

existing corporate portfolio, or the market as a whole.

If we now relax our

all-equity

assumption to introduce an element of cheaper borrowing (debt) into the

corporate ˚nancial mix, managerial policies designed to maximise shareholder wealth comprise two

distinct but nevertheless

inter-related

functions.

Ł ˜e

investment function

, which identi˚es and selects a portfolio of investment opportunities

that

maximise

expected

net cash in˛ows

(ENPV) commensurate with risk.

Ł ˜e ˝nance function

, which identi˚es potential fund sources (equity and debt, long or short)

required to sustain investments.

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15 An Overview

Management™s task now extends beyond satisfying shareholder expectations. ˜ey need to evaluate the

risk-adjusted

return for each capital source. ˜en select the optimum structure that will

minimise

their

overall weighted average cost of capital (WACC) as a discount rate for project appraisal. However, the

principles of investment still apply.

Figure 1:2:

Corporate Economic Performance Œ Winners and Losers.

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16 An Overview

Figure 1.2 distinguishes the ﬁwinnersﬂ from the ﬁlosersﬂ in their drive to create wealth by summarising in

˚nancial terms why some companies fail. ˜ese may then fall prey to take-over as share values plummet,

or even become bankrupt and disappear altogether.

Ł Companies engaged in ineˆcient or irrelevant activities, which produce losses (negative ENPV)

are gradually starved of ˚nance because of reduced dividends, inadequate retentions and

the capital market™s unwillingness to replenish their borrowing, thereby producing a fall in

share price.

FINANCE Acquisition Disposition INVESTMENT Equity of Funds of Funds Fixed Assets Retentions Current Assets Debt Current Liabilities Objective Objective Minimum Maximum Cash Cost (WACC) Profit (ENPV) Finance Function Investment Function Objective Maximum Share Price Figure 1.3:

Corporate Financial Objectives

Figure 1.3 summarises the strategic objectives of ˚nancial management relative to the inter-relationship

between

internal

investment and

external

˚nance decisions that enhance shareholder wealth (share price)

based on the law of supply and demand to attract more rational-risk averse investors to the company.

˜e diagram reveals that a company wishing to maximise its wealth using share price as a

vehicle

, must

create cash pro˚ts using ENPV as the

driver

. Management would not wish to invest funds in capital

projects unless their

marginal

yield at least matched the rate of return prospective investors can earn

elsewhere on comparable investments of equivalent risk.

In an ideal world, total cash pro˜ts from a portfolio of investments should exceed the overall cost of investment (WACC)

producing a positive ENPV, which not only covers all interest on debt but also yields a residual that satis˜es shareholder

expectations, to be either distributed as a dividend, or retained to ˜nance future pro˜table investments.

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17 An Overview

1.3 Questions to Consider

So far so good: but what if capital markets are

imperfect

?Information is not freely available and there are barriers to trade. Moreover, if a signi˚cant number

of market participants, including corporate management, ˚nancial institutions and private investors,

pursue their own agenda, characterised by short-term goals at the expense of long-run shareholder

wealth maximisation?

Ł Are shares still correctly priced?

Ł Are ˚nancial resources still allocated to the most pro˚table investment opportunities,

irrespective of shareholder consumption preferences?

In other words, are markets

e˚cient

once the

agency

principle breaks down and short-termism takes hold?

As all other texts in my

bookboon

series suggest, based on historical real-world volatility mentioned

earlier, perhaps they are not.

Post-modern

theorists with cutting-edge mathematical expositions of ﬁspeculativeﬂ bubbles, ﬁcatastropheﬂ

theory and market ﬁincoherenceﬂ, now hypothesise that classical statistical analyses (Quants) are

discredited. Investment prices and returns may be

non-random

variables and markets

have a memory

. ˜is ﬁnew ˚nanceﬂ takes a

non-linear

view of society, which frequently dispenses with the assumption

that we can

maximise

anything.

Unfortunately, none of these models are yet suˆciently re˚ned to provide market participants with

alternative guidance in their quest for greater wealth. ˜is explains why the investment community

still clings to the time-honoured objective of shareholder wealth maximisation, based on Quants as a

framework for analysis.

Nevertheless,

post-modernism

serves a dual theoretical purpose mentioned at the outset.

Ł First, it reminds us that the foundations of traditional

modern

˚nance may sometimes be ﬁbad

scienceﬂ by which we mean that theoretical investment and ˚nancing decisions are all too o˝en

based on simplifying assumptions without any empirical support.

Ł Second, it reveals why investors (sophisticated or otherwise) should always interpret

conventional statistical analyses of wealth maximisation behaviour with caution and not be

surprised if subsequent events invalidate their conclusions.

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18 An Overview

1.4 Fisher™s Legacy and Modigliani-Miller

Once a company has made an issue of ordinary shares and received the proceeds, management is neither

directly involved with their subsequent transactions on the capital market, nor the prices at which they are

transacted. ˜ese are matters of negotiation between prevailing shareholders and prospective investors.

In sophisticated, mixed market economies where

ownership is divorced from control

, the normative objective of modern

˜nancial management is therefore de˜ned by the maximisation of shareholder wealth, based on ENPV maximisation

using mean-variance analysis.

We examined these propositions by considering perfect (eˆcient) capital markets under conditions of

certainty with

no barriers to trade

, characterised by freedom of information, no transaction costs and

tax neutrality. According to Fisher™s Separation ˜eorem, Jenson and Meckling™s Agency ˜eory and

the EMH of Fama (

op.cit

):An

all-equity

˜rm can justify retained earnings to ˜nance future investments, rather than pay a current dividend, if their

marginal return on new projects at least equals the market rate of interest that shareholders could obtain by using

dividends to ˜nance alternative investments of equivalent business risk elsewhere.

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19 An Overview

Even if markets are uncertain, providing they are still eˆcient, rational, risk-averse shareholders should

support such behaviour. It cannot detract from their wealth, because at any point in time, retentions

and dividends are perceived as

perfect economic substitutes

. What they lose through dividends foregone,

they expect to receive through increased equity value (capital gains) generated by internally ˚nanced

projects discounted at their required opportunity rate of return.

And this is where MM ˚rst contribute to our analysis.

According to their

dividend irrelevancy

hypothesis (1961) explained in Chapter Four, when shareholders

need to replace a missing dividend to satisfy their consumption preferences, the solution is simple.

Ł Shareholders can create a

home-made

dividend by either borrowing an equivalent amount at

the same rate as the company, or sell shares at a price that re˙ects their earnings and reap the

capital gain.

Since the borrowing (discount) rate is entirely determined by the

business

risk of investment (the

variability of future earnings) and not

˝nancial

risk (the pattern of dividends), the ˚rm™s distribution

policy is trivial.

Ł Dividend decisions are concerned with what is done with earnings but do not determine the

risk originally associated with the quality of investment that produces them.

To set the scene for MM, let us therefore consider a simple example that clari˚es the inter-relationship

between shareholder wealth maximisation, the

supremacy

of investment policy and the

irrelevance

of

dividend (˚nancial) policy, given the assumptions of a perfect market.

Review Activity

Suppose a company has issued ordinary shares (common stock) which generate a net annual cash˚ow of £1 million

in perpetuity to be paid out as dividends. The market rate of interest and corporate discount rate commensurate with

the degree of risk is 10 percent.

The capitalisation of this constant dividend stream (a formula with which you should be familiar) de˜nes a total

equity value:

VE = £1 million / 0.10

= £10 millionThe company now intends to ˜nance a new project of equivalent risk by retaining the next dividend to generate a net

cash in˚ow of £2 million twelve months later, paid out as an additional dividend. Thereafter a full distribution policy

will be adhered to.

Required:

Is management correct to retain earnings and would you invest in the company?

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20 An Overview

An Indicative Outline Solution

˜e data provides an opportunity to review your knowledge of the investment and ˚nancial criteria that

underpin the normative objective of shareholder wealth maximisation, using NPV maximisation as a

determinant of share price.

Ł ˜e Optimum Dividend-Retention Policy

˜e ˚rst question we must ask ourselves is whether the incremental investment ˚nanced by the non-

payment of a dividend a˛ects the shareholders adversely?

We can present the managerial decision in terms of the revised dividend stream:

t0 t1 t2 t3–.t £ million £ £ £ £ £Existing dividends 1 1 1 1Project cash ˚ows

(1) 2 -Revised dividends

- 3 1 1If we now compare total equity value using the

discounted value

of future dividends:

VE (existing) = £1 million / 0.10

= £10 million

VE (revised) = £3 million / (1.1)

2 + (£1 million / 0.10) / (1.1)

2 = £10.744 million

Once the project is accepted, the present value (PV) of the ˚rm™s equity capital will rise and the

shareholders will be £744,000 better o˛ with a revised dividend stream.

Perhaps you need to pause here, because the application of the discounted cash˙ow (DCF) formula to

the new valuation of the dividends requires explanation. If so, take time out to revise your understanding

of its rationale before we proceed.

Ł Net Present Value (NPV) Maximisation

If you are comfortable with DCF analysis, we can determine the same wealth maximisation decision

without even considering the fact that the pattern of dividends has changed, thereby proving the veracity

of Fisher™s Separation ˜eorem and the MM dividend irrelevancy hypothesis quite independently.

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21 An Overview

˜e increase in total value is simply the new project™s

net present value

(NPV). ˜is is proven by

implementing the corporate DCF capital budgeting model, with which you are familiar:

NPV = (£1million)

+ £2 million

= £744,000 1.1 (1.1)2In our example, the shareholders simply relinquish their next dividend and gain an increase in the

subsequent value of their ordinary shares from £10,000,000 to £10,744,000.

Conclusions

Ł In a perfect capital market, where the ˚rm™s investment decisions can be made independently of

the consumption decisions of shareholders, NPV project maximisation represents shareholder

wealth maximising behaviour.

Ł It is the

investment

decision that has determined the value of equity and

not

the

˝nancing

(dividend) decision.

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22 An Overview

1.5 Summary and Conclusions

˜e remainder of this study is designed to complement and develop your understanding of the normative

shareholder wealth maximisation objective, within the context of modern ˚nance theory and MM™s

pivotal ﬁlaw of one priceﬂ.

It extends beyond all-equity ˚rms, dominated by the

irrelevance

of dividend policy relative to corporate

value, into a world of corporate borrowing (leverage) and a multiplicity (portfolio) of investments. And

as we shall discover, MM™s basic position is entirely consistent.

The overall cut-o˛ rate for investment and corporate value are

independent

of ˜nancial structure. Just like dividend-

retention policies, companies agonising over whether to issue debt or equity are wasting their time.

Like my previous

bookboon

texts, some topics will focus on ˚nancial numeracy and mathematical

modelling. Others will require a literary approach. ˜e rationale is to vary the pace and style of the

learning experience. It not only applies mathematics and accounting formulae through a series of

Activities (with outline solutions) some of which are sequential, but also develops your own arguments

and a critique of the subject as a guide to further study.

1.6 Selected References

Hill, R.A.,

bookboon.com

.Text Books:

Strategic Financial Management

, 2008.Strategic Financial Management: Exercises,

2009.Portfolio ˜eory and Financial Analyses, 2010.

Portfolio ˜eory and Financial Analyses: Exercises, 2010.

Corporate Valuation and Takeover, 2011.

Corporate Valuation and Takeover: Exercises, 2012.

Working Capital and Strategic Debtor Management,

2013.Working Capital and Debtor Management: Exercises

2013.Download free eBooks at bookboon.com

23 An Overview

Business Texts:

Strategic Financial Management: Part I, 2010.

Strategic Financial Management: Part II, 2010.

Portfolio ˜eory and Investment Analysis,

2010.˜e Capital Asset Pricing Model, 2010.

Company Valuation and Share Price, 2012.

Company Valuation and Takeover, 2012.

Working Capital Management: ˜eory and Strategy, 2013.

Strategic Debtor Management and the Terms of Sale, 2013.

Portfolio ˜eory and Investment Analysis, 2

nd Edition,

2014.˜e Capital Asset Pricing Model, 2010, 2

nd Edition, 2014.

1. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

2. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,

Journal

of Business of the University of Chicago,

Vol. 34, No. 4, October 1961.

3. Fisher, I.,

˙e ˙eory of Interest

, Macmillan, 1930.

4. Jensen, M.C. and Meckling, W.H., ﬁ˜eory of the Firm: Managerial Behaviour, Agency Costs

and Ownership Structureﬂ,

Journal of Financial Economics

, 3, October 1976.

5. Fama, E.F., ﬁ˜e Behaviour of Stock Market Pricesﬂ,

Journal of Business

, Vol. 38, 1965.

6. Markowitz, H.M., ﬁPortfolio Selectionﬂ,

Journal of Finance

, Vol. 13, No. 1, 1952.

7. Tobin, J., ﬁLiquidity Preferences as Behaviour Towards Riskﬂ, Review of Economic Studies,

February 1958.

8. Sharpe, W., ﬁA Simpli˚ed Model for Portfolio Analysisﬂ,

Management Science,

Vol. 9, No. 2,

January 1963.

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24 Part Two:

The Dividend Decision

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25 How to Value a Share

2

How to Value a Share

Introduction

Part One

surveyed the development of modern ˚nance theory, based on Fisher™s Separation ˜eorem

(1930) with speci˚c reference to the

Investment Decision

, to illustrate why a preponderance of academics

and analysts still support the

normative

objective of shareholder wealth maximisation. Based on

management™s expected NPV (ENPV) maximisation of all a ˚rm™s projects and its impact on the market

price of equity, we explained how under certain conditions:

An

all-equity

˜rm can justify retained earnings to ˜nance future investments, rather than pay a current dividend, if their

marginal return on new projects at least equals the market rate of interest that shareholders could obtain by using

dividends to ˜nance alternative investments of equivalent business risk elsewhere.

Even if markets are uncertain, providing they are still eˆcient, rational, risk-averse shareholders should

support such behaviour. It cannot detract from their wealth, because at any point in time, retentions

and dividends are perceived as

perfect economic substitutes

. What they lose through dividends foregone,

they expect to receive through increased equity value (capital gains) generated by internally ˚nanced

projects discounted at their required opportunity rate of return.

Download free eBooks at bookboon.com

26 How to Value a Share

So far, so good: throughout Part One we accepted without question the fundamental assumption that

dividends and earnings are

equally

valued by investors who model share price. However:

Ł If dividends and retentions are not

perfect economic substitutes

, a ˚rm™s distribution policy may

determine an optimum share price and hence share price maximisation, which runs counter

to the ﬁdividend irrelevancyﬂ hypothesis of Miller and Modigliani (1961).

Part Two

now deals explicitly with MM and the

Dividend Decision

, namely its impact on current share

price and the market capitalisation of equity (

i.e. shareholders™ wealth) determined by the consequence

of managerial

˝nancial

policies to distribute or retain pro˚ts, which stem from their

investment

decisions.

˜e key to understanding stock market performance, used by investors to analyse these inter-relationships,

requires a theoretical appreciation of the relationship between a share™s value and its return (dividend or

earnings) using various models based on discounted revenue theory.

To set the scene, we shall keep this Chapter™s analysis simple by outlining the theoretical determinants

of share price, with particular reference to the

capitalisation of a perpetual annuity

using both dividend

and earnings yield formulae.

Detailed consideration of the MM controversy as to whether dividends or earnings are a prime determinant

of share value will then be covered in subsequent Chapters, with reference to their comprehensive critique

of the case for dividends presented by Myron J. Gordon (1962).

Ł According MM™s ﬁlaw of one priceﬂ the current value of an all-equity ˚rm is

dependent

upon

its investment strategy and

independent

of its dividend policy.

Ł ˜e variability of earnings, (

business

risk) rather than how they are packaged for distribution

(˝nancial

risk) determines the shareholders™ desired rate of return (cost of equity) and

management™s cut-o˛ rate for investment (project discount rate) and hence its share price.

Part ˜ree

(the

Finance Decision

) then introduces MM™s entirely consistent theory of capital structure

by relaxing our

all-equity

assumption to introduce an element of cheaper borrowing (debt) into the

corporate ˚nancial mix, premised on managerial policies designed to maximise shareholder wealth.

By reformulating the share valuation models of Part Two and introducing the pricing and return of

loan stock and other sources of ˚nance, a managerial cut-o˛ rate for project appraisal using an overall

weighted average cost of capital (WACC) will be derived. Given its assumptions and limitations, we shall

then consider the vexed question as to whether capital gearing (leverage) is a

determinant

of WACC and

total corporate value (the ﬁtraditionalﬂ view) or an

irrelevance

as MM hypothesise.

Download free eBooks at bookboon.com

27 How to Value a Share

Based on their

arbitrage

concept (1958) we shall arrive at two conclusions, which conform to MM™s

dividend irrelevancy position.

Ł Total corporate value (debt plus equity) represented by the expected NPV of a ˚rm™s income

stream discounted at a rate appropriate to its business risk, should be una˛ected by ˚nancial

risk associated with its mode of ˚nancing.

Ł Any rational debt-equity ratio should produce the same overall cut-o˛ rate for investment

(WACC) equivalent to the cost of equity in an all-equity ˚rm.

Part Four

(the

Portfolio Decision

) establishes a ˚nal mathematical connection between MM™s ﬁlaw of one

priceﬂ and Modern Portfolio ˜eory (MPT), with speci˚c reference to the general Capital Asset Pricing

Model (CAPM) of William Sharpe (1963).

According to the CAPM and

beta

factor

analysis, if di˛erent capital projects are combined into an

optimum portfolio, management can plot an ﬁeˆciency frontierﬂ using Quants analysis and then select

further investments for inclusion into the existing asset mix, according to their desired risk-return pro˚le

(utility curve).

As we shall discover, without debt in it capital structure, a company™s

asset

beta equals its

equity

beta

for projects of equivalent business risk. However, according to MM™s theory of capital structure and the

arbitrage

process:

Ł Companies that are identical in every respect apart from their gearing should also have

identical asset beta factors because the variability of earnings is the same. ˜ese factors are

not in˙uenced by ˚nancial risk.

Ł So, just like WACC (relative to the cost of equity in an unlevered ˚rm) the asset beta (equity

beta) of an all-equity company can be used to evaluate geared projects in the same class of

business risk without considering di˛erences in ˚nancial structure.

2.1 The Capitalisation Concept

Discounted revenue theory de˚nes an investment™s present value (PV) as the sum of its relevant periodic

cash ˙ows (C

t) discounted at an appropriate opportunity cost of capital, or rate of return (r) on alternative

investments of equivalent risk over time (n). Expressed algebraically:

1. The World of Modigliani and Miller

27 How to Value a Share

Based on their

arbitrage

concept (1958) we shall arrive at two conclusions, which conform to MM™s

dividend irrelevancy position.

Ł Total corporate value (debt plus equity) represented by the expected NPV of a ˜rm™s income

stream discounted at a rate appropriate to its business risk, should be una˚ected by ˜nancial

risk associated with its mode of ˜nancing.

Ł Any rational debt-equity ratio should produce the `same overall cut-o˚ rate for investment

(WACC) equivalent to the cost of equity in an all-equity ˜rm.

Part Four

(the

Portfolio Decision

) establishes a ˜nal mathematical connection between MM™s ﬁlaw of one

priceﬂ and Modern Portfolio ˛eory (MPT), with speci˜c reference to the general Capital Asset Pricing

Model (CAPM) of William Sharpe (1963).

According to the CAPM and

beta

factor

analysis, if di˚erent capital projects are combined into an

optimum portfolio, management can plot an ﬁe˝ciency frontierﬂ using Quants analysis and then select

further investments for inclusion into the existing asset mix, according to their desired risk-return pro˜le

(utility curve).

As we shall discover, without debt in it capital structure, a company™s

asset

beta equals its

equity

beta

for projects of equivalent business risk. However, according to MM™s theory of capital structure and the

arbitrage

process:

Ł Companies that are identical in every respect apart from their gearing should also have

identical asset beta factors because the variability of earnings is the same. ˛ese factors are

not in˙uenced by ˜nancial risk.

Ł So, just like WACC (relative to the cost of equity in an unlevered ˜rm) the asset beta (equity

beta) of an all-equity company can be used to evaluate geared projects in the same class of

business risk without considering di˚erences in ˜nancial structure.

2.1 The Capitalisation Concept

Discounted revenue theory de˜nes an investment™s present value (PV) as the sum of its relevant periodic

cash ˙ows (C

t) discounted at an appropriate opportunity cost of capital, or rate of return (r) on alternative

investments of equivalent risk over time (n). Expressed algebraically:

1. n PVn = Ct /(1+r) t t=1 ˛e equation has a convenient property. If the investment™s annual return (r) and cash receipts (C

t) are

constant and tend to in˜nity

, (Ct = C1 = C2 = C3 = C ) their PV simpli˜es to the formula for the

capitalisation of a constant perpetual annuity

:2. PV = Ct / r = C1 / r ˜e equation has a convenient property. If the investment™s annual return (r) and cash receipts (C

t) are

constant and tend to in˝nity

, (Ct = C1 = C2 = C3 = Cˇ) their PV simpli˚es to the formula for the

capitalisation of a constant perpetual annuity

:2. PVˇ = Ct / r C1 / rDownload free eBooks at bookboon.com

28 How to Value a Share

˜e return term (r) is called the

capitalisation

rate because the transformation of a cash ˙ow series

into a capital value (PV) is termed ﬁcapitalisationﬂ. With data on PV

ˇ and r, or PV

ˇ and C

t, we can also

determine C

t and r respectively. Rearranging Equation (2) with one unknown:

3. Ct = PVˇ x r4. r = PVˇ / CtActivity 1

The previous PV equations are vital to your understanding of the various share valuation models that follow. If you are

unsure of their theory and application, then I recommend that you download

Strategic Financial Management

(SFM

) from the author™s

bookboon

series and read Chapters Two and Five before you continue.

Having completed this reading, you will also appreciate that shares may be traded either

cum-div

or

ex-div

, which means they either include (cumulate) or exclude the latest dividend. For example, if you sell a

share

cum-div

today for P

0 the investor also receives the current dividend D

0. Excluding any transaction

costs, the investor therefore pays a total price of (D

0 + P0). Sold

ex -div

you would retain the dividend.

So, the trade is only based on current price (P

0). Download free eBooks at bookboon.com

29 How to Value a Share

˜is distinction between

cum-div

and

ex-div

is important throughout the remainder of our study because

unless speci˚ed otherwise, we shall adopt the time-honoured academic convention of de˚ning the current

price of a share using an

ex-div

valuation.

2.2 The Capitalisation of Dividends and Earnings

Irrespective of whether shares are traded

cum-div or ex-div

, their present values can be modelled in a

variety

of ways using discounted revenue theory. Each depends on a de˚nition of future periodic income

(either a dividend or earnings stream) and an appropriate discount rate (either a dividend or earnings

yield) also termed the equity capitalisation rate.

For example, given a forecast of periodic future dividends (D

t) and a shareholder™s desired rate of return

(Ke) based on current dividend yields for similar companies of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) should equal the discounted sum

of future dividends (D

t) plus its eventual

ex-div

sale price (P

n) using the current dividend yield (K

e) as a capitalisation rate

Expressed algebraically:

5. P0 = [(D1 /1 + Ke) + (D2 /1 + Ke)2 + – + (Dn /1 + Ke) n] + (Pn /1 + Ke) nRewritten and simpli˚ed, this de˚nes the

˝nite-period dividend valuation model:

6. The World of Modigliani and Miller

29 How to Value a Share

˜is distinction between

cum-div

and

ex-div

is important throughout the remainder of our study because

unless speci˚ed otherwise, we shall adopt the time-honoured academic convention of de˚ning the current

price of a share using an

ex-div

valuation.

2.2 The Capitalisation of Dividends and Earnings

Irrespective of whether shares are traded

cum-div or ex-div

, their present values can be modelled in a

variety

of ways using discounted revenue theory. Each depends on a de˚nition of future periodic income

(either a dividend or earnings stream) and an appropriate discount rate (either a dividend or earnings

yield) also termed the equity capitalisation rate.

For example, given a forecast of periodic future dividends (D

t) and a shareholder™s desired rate of return

(Ke) based on current dividend yields for similar companies of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) should equal the discounted sum

of future dividends (D

t) plus its eventual

ex-div

sale price (P

n) using the current dividend yield (K

e) as a capitalisation rate

Expressed algebraically:

5. P0 = [(D1 /1 + Ke) + (D2 /1 + Ke)2 + – + (Dn /1 + Ke) n] + (Pn /1 + Ke) nRewritten and simpli˚ed, this de˚nes the

˜nite-period dividend valuation model:

6. n P0 = Dt /(1+Ke) t + Pn /(1 + Ke) n t=1 Likewise, given a forecast for periodic future earnings (E

t) and a desired return (K

e) based on current

earnings yields of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) equals the sum of future earnings (E

t) plus its eventual

ex-div

sale price (P

n) all discounted at the current earnings yield (K

e).Algebraically, this de˚nes the

˜nite-period earnings valuation model:

7. n P0 = Et /(1+Ke) t + Pn /(1 + Ke) n t=1 Pn /(1 + Ke) nLikewise, given a forecast for periodic future earnings (E

t) and a desired return (K

e) based on current

earnings yields of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) equals the sum of future earnings (E

t) plus its eventual

ex-div

sale price (P

n) all discounted at the current earnings yield (K

e).Algebraically, this de˚nes the

˝nite-period earnings valuation model:

7. The World of Modigliani and Miller

29 How to Value a Share

˜is distinction between

cum-div

and

ex-div

is important throughout the remainder of our study because

unless speci˚ed otherwise, we shall adopt the time-honoured academic convention of de˚ning the current

price of a share using an

ex-div

valuation.

2.2 The Capitalisation of Dividends and Earnings

Irrespective of whether shares are traded

cum-div or ex-div

, their present values can be modelled in a

variety

of ways using discounted revenue theory. Each depends on a de˚nition of future periodic income

(either a dividend or earnings stream) and an appropriate discount rate (either a dividend or earnings

yield) also termed the equity capitalisation rate.

For example, given a forecast of periodic future dividends (D

t) and a shareholder™s desired rate of return

(Ke) based on current dividend yields for similar companies of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) should equal the discounted sum

of future dividends (D

t) plus its eventual

ex-div

sale price (P

n) using the current dividend yield (K

e) as a capitalisation rate

Expressed algebraically:

5. P0 = [(D1 /1 + Ke) + (D2 /1 + Ke)2 + – + (Dn /1 + Ke) n] + (Pn /1 + Ke) nRewritten and simpli˚ed, this de˚nes the

˜nite-period dividend valuation model:

6. n P0 = Dt /(1+Ke) t + Pn /(1 + Ke) n t=1 Likewise, given a forecast for periodic future earnings (E

t) and a desired return (K

e) based on current

earnings yields of equivalent risk:

The present

ex-div

value (P

0) of a share held for a

given

number of years (n) equals the sum of future earnings (E

t) plus its eventual

ex-div

sale price (P

n) all discounted at the current earnings yield (K

e).Algebraically, this de˚nes the

˜nite-period earnings valuation model:

7. n P0 = Et /(1+Ke) t + Pn /(1 + Ke) n t=1 Pn /(1 + Ke) nDownload free eBooks at bookboon.com

30 How to Value a Share

Activity 2

We observed in Part One that a logical approach to ˜nancial analysis is to make

simplifying assumptions that rationalise

its complexity

. A classic example is the derivation of a series of dividend and earnings valuations, other than the

˜nite

model. Some are more sophisticated than others, but their common purpose is to enable investors to assess a share™s

performance under a variety of conditions.

To illustrate the point, brie˚y summarise the theoretical assumptions

and de˜nitions for the following models based on

your reading of

SFM

(Chapter Five) or any other source material.

The

single-period

dividend valuation

The

general

dividend valuation

The

constant

dividend valuation

Then give some thought to which of these models underpins the data contained in stock exchange listings published

by the ˜nancial press worldwide.

We know that the

˝nite-period

dividend valuation model assumes that a share is held for a given number

of years (n). So, today™s

ex div

value equals a series of expected year-end dividends (D

t) plus the expected

ex-div

price at the end of the entire period (P

n), all discounted at an appropriate dividend yield (K

e) for

shares in that risk class. Adapting this formulation we can therefore de˚ne:

-˜e

single-period

dividend valuation model

Assume you hold a share for one period (say a year) at the end of which a dividend is paid. Its

current

ex div

value is given by the expected year-end dividend (D

1) plus an

ex-div

price (P

1) discounted at an appropriate dividend yield (K

e). -˜e general

dividend valuation model

If a share is held inde˚nitely, its current

ex div

value is given by the summation of an in˚nite

series of year-end dividends (D

t) discounted at an appropriate dividend yield (K

e). Because the

share is never sold, there is no ˚nal

ex-div

term in the equation.

-˜e

constant

dividend valuation model

If the annual dividend (D

t) not only tends to in˚nity but also remains constant, and the current

yield (K

e) doesn™t change, then the

general

dividend model further simpli˚es to the

capitalisation

of a perpetual annuity

.Download free eBooks at bookboon.com

31 How to Value a Share

2.3 The Capitalisation of Current Maintainable Yield

Your answers to Activity 2 not only reveal the impact of di˛erent assumptions on a share™s theoretical

present value, but why basic price and yield data contained in stock exchanges listings published by the

˚nancial press and internet favour the

constant

valuation model, rather than any other.

˜ink about it. ˜e derivation and analyses of current share prices based on future estimates of dividends,

ex-div

prices and appropriate discount rates for billions of market participants, even over a single period

is an impossible task.

To avoid any forecasting weakness, characterised by uncertainty and to provide a

benchmark

valuation for

the greatest possible number, stock exchange listings therefore assume that shares are held in

perpetuity

and the latest reported dividend per share will remain

constant

over time. ˜is still allows individual

investors with other preferences, or information to the contrary, to model more complex assumptions

for comparison. ˜ere is also the added commercial advantage that by using simple metrics, newspaper

and internet stock exchange listings should have universal appeal for the widest possible readership.

Turning to the mathematics, given your knowledge of discounted revenue theory and the

capitalisation

of a perpetual annuity

(where PV = C

t / r) share price listings de˚ne a current

ex- div

price (P

0) using

the

constant

dividend valuation model as follows:

8. P0 = D1 / KeDownload free eBooks at bookboon.com

32 How to Value a Share

Next year™s dividend (D

1) and those therea˝er are represented by the latest reported dividend (

i.e. a constant). Rearranging terms, (K

e) the shareholders desired rate of return (equity capitalisation rate) is

also a constant represented by the current yield, which is assumed to be

maintainable

inde˚nitely.

9. Ke = D1 / P02.4 The Capitalisation of Earnings

For the purpose of exposition, so far we have focussed on dividend income as a determinant of price

and value, with only passing reference to earnings. But what about shareholders interested in their

total

periodic returns (dividends plus retentions) from corporate investment? ˜ey need to capitalise a post-

tax earnings stream (E

t) such as

earnings per share

(EPS) and analyse its yield (K

e). No problem: the

structure

of the valuation models summarised in Activity 2 remains the same but E

t is substituted for

Dt and K

e now represents an earnings yield, rather than a dividend yield. ˜us, we can de˚ne a parallel

series of equations using:

˜e

single-period

, earnings valuation model

˜e

˝nite-period

, earnings valuation model

˜e

general

earnings valuation model

˜e

constant

earnings valuation model

Turning to stock exchange listings, the ˚nancial press and internet, we also observe that for simplicity

the publication of earnings data is still based on the

capitalisation of a perpetual annuity

.10. P0 = E1 / KeNext year™s earnings (E

1) and those therea˝er are represented by the latest reported pro˚t (

i.e. a constant).

Rearranging terms, (K

e) the shareholders desired rate of return (equity capitalisation rate) is also a

constant represented by the current earnings yield, which is assumed to be

maintainable

inde˚nitely.

11. Ke = E1 / P0Review Activity

Having downloaded this text and perhaps others in my

bookboon

series, it is reasonable to assume that you can already

interpret a set of published ˜nancial accounts and share price data. To test your level of understanding for future reference,

select a newspaper of your choice and a number of companies from its stock exchange listings. Then use the data:

1. To explain the mathematical relationship between a company™s dividend and earnings yields and why the two

may di˛er.

2. To de˜ne earnings yields published in the ˜nancial press.

Download free eBooks at bookboon.com

33 How to Value a Share

An Indicative Outline Solution

1. ˜e Mathematical Yield Relationship

Our discussion of eˆcient markets in Chapter One explained why a company™s shares cannot sell for

di˛erent prices at a particular point in time. So, it follows that:

12. P0 = D1 / Ke = E1 / Ke And if a company adopts a policy of full distribution (whereby D

1 = E1) then the equity capitalisation

rates for dividends and earnings, using a current maintainable yield (K

e) must also be identical.

13. Ke = D1 / P0 = Ke = E1 / P0 But what about the more usual situation, where a company retains a proportion of earnings for

reinvestment? By de˚nition, the respective equity capitalisation rates (K

e) must now di˛er because

14. Ke = D1 / P0 < Ke = E1 / P0 Given P

0 and D

1 < E1As we shall discover in Chapter ˜ree, there is a

behavioural

explanation for the relationship between the

two yields. For the moment, suˆce it to say that there is also an underlying

mathematical

relationship.

For example, if a company™s current share price, latest reported dividend and earnings per share are

$100, $10 and $20 respectively, then because earnings

cover

dividends twice the dividend yield is half

the earnings yield (10 and 20 percent respectively).

˜is di˛erence in yields is not a problem for investors who know what they are looking for. Some will

prefer their return as current income (dividends and perhaps the sale of shares). Some will look to earnings

that incorporate retentions (future dividends plus capital gains). Most will hedge their bets by combining

the two in share portfolios that minimise risk. So, their respective returns will di˛er according to their

risk-return pro˚le. Which is why share price listings in newspapers worldwide focus on dividends

and

earnings, as well as the

interrelationship

between the two measured by dividend cover.

2. ˜e Yield and Price-Earnings (P/E) Ratio

Moving on to the second question posed by our Review Activity, if you are at all familiar with share price

listings published in the ˚nancial press, you will be aware of a

convention

that also enables investors to

avoid any confusion between

dividend and earnings yields when analysing a share™s performance.

Given the current earnings yield:

11. Ke = E1 / P0Download free eBooks at bookboon.com

34 How to Value a Share

˜e equation™s terms can be rearranged to produce its

reciprocal

, the price-earnings (P/E) ratio.

15. P/E = P0 / E1 = 1/KeUnlike the earnings yield, which is a

percentage

return,

the P/E ratio is a

real

number that analyses price as a

multiple

of earnings. On the assumption that a ˜rm™s current post tax pro˜ts are maintainable inde˜nitely, the ratio therefore provides

an alternative method whereby a company™s distributable earnings can be capitalised to establish a share™s value.

Because the two measures are

reciprocals

whose product always equals one, the interpretation of the

P/E is that the

lower

the number, the

higher

the earnings yield and

vice versa

. And because investors

are dealing with an

absolute

P/E value and not a

percentage

yield, there is no possibility of confusing a

share™s dividend and earnings performance when reading share price listings, articles or commentaries

from the press, media, analyst reports, or internet downloads.

Finally, having noted that low valuation multipliers correspond to high

returns and that a number

multiplied by its reciprocal equal™s one: use

Table 2.1 to con˚rm a

perfect inverse

relationship between a

share™s P/E and its earnings yield. Not only will this exercise be useful for future reference throughout

this text, but your future reading of the ˚nancial press should also fall into place.

Download free eBooks at bookboon.com

35 How to Value a Share

The World of Modigliani and Miller

35 How to Value a Share

Table 2.1:

The Relationship between the P/E Ratio and Earnings Yields

2.5 Summary and Conclusions

˜is Chapter has outlined the fundamental relationships between share valuation models and the

derivation of the cost of equity capital for the purpose of analysing stock market returns.

We set the scene by explaining the derivation of share valuation models using discounted revenue theory,

with reference to the capitalisation of a perpetual annuity. We noted that corresponding equity valuations

based on current dividend and earnings should be ˚nancially equivalent.

˜e relationship between an

ex-div

dividend and earnings valuation revealed why a few select metrics

(based on price, dividend yield, the P/E ratio and cover) published in the media encapsulate a company™s

stock market performance and provide a guide to future investment.

However, as we shall discover in later chapters, a share™s intrinsic value (price) is only meaningful if we

move beyond the mathematics and place it in a

behavioural

context. For example, given a company™s latest

reported dividend and pro˚t ˚gures, investors can use existing dividend yields and P/E ratios to place a

comparative value on that company™s shares. ˜ese can then be compared with its actual value (current

market price) to establish whether the company is either undervalued, equitable, or overvalued, relative

to the market for similar shares of equivalent risk. Needless to say, undervalued, rational investors buy,

equitable they hold, overvalued they sell.

But what motivates their trading decisions: is it the dividend policy of the ˚rm, or its earning potential?

2.6 Selected References

1. Fisher, I.,

˜e ˜eory of Interest

, Macmillan, 1930.

2. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,

Journal

of Business of the University of Chicago,

Vol. 34, No. 4, October 1961.

3. Gordon, M.J.,

˜e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

4. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

5. Sharpe, W., ﬁA Simpli˚ed Model for Portfolio Analysisﬂ,

Management Science,

Vol. 9, No. 2,

January 1963.

6. Hill, R.A.,

Strategic Financial Management:

Chapters Two and Five,

bookboon.com

, 2008.Table 2.1:

The Relationship between the P/E Ratio and Earnings Yields

2.5 Summary and Conclusions

˜is Chapter has outlined the fundamental relationships between share valuation models and the

derivation of the cost of equity capital for the purpose of analysing stock market returns.

We set the scene by explaining the derivation of share valuation models using discounted revenue theory,

with reference to the capitalisation of a perpetual annuity. We noted that corresponding equity valuations

based on current dividend and earnings should be ˚nancially equivalent.

˜e relationship between an

ex-div

dividend and earnings valuation revealed why a few select metrics

(based on price, dividend yield, the P/E ratio and cover) published in the media encapsulate a company™s

stock market performance and provide a guide to future investment.

However, as we shall discover in later chapters, a share™s intrinsic value (price) is only meaningful if we

move beyond the mathematics and place it in a

behavioural

context. For example, given a company™s latest

reported dividend and pro˚t ˚gures, investors can use existing dividend yields and P/E ratios to place a

comparative value on that company™s shares. ˜ese can then be compared with its actual value (current

market price) to establish whether the company is either undervalued, equitable, or overvalued, relative

to the market for similar shares of equivalent risk. Needless to say, undervalued, rational investors buy,

equitable they hold, overvalued they sell.

But what motivates their trading decisions: is it the dividend policy of the ˚rm, or its earning potential?

2.6 Selected References

1. Fisher, I.,

˙e ˙eory of Interest

, Macmillan, 1930.

2. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,

Journal

of Business of the University of Chicago,

Vol. 34, No. 4, October 1961.

3. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

4. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

5. Sharpe, W., ﬁA Simpli˚ed Model for Portfolio Analysisﬂ,

Management Science,

Vol. 9, No. 2,

January 1963.

6. Hill, R.A.,

Strategic Financial Management:

Chapters Two and Five,

bookboon.com

, 2008.Download free eBooks at bookboon.com

36 The Role of Dividend Policy

3 The Role of Dividend Policy

Introduction

For simplicity, we have assumed that if shares are held inde˚nitely and future dividends or earnings per

share remain constant, their current

ex-div

price can be expressed using the

capitalisation of a perpetual

annuity

based on current dividend or earnings yields. ˜is Chapter re˚nes these

constant

valuation

models by considering two inter-related questions.

-What happens to current share price if forecast dividends or earnings are not constant in

perpetuity?

-When valuing a company™s shares, do investors value current dividends more highly than

earnings retained for future investment?

3.1 The Gordon Growth Model

Chapter One began with a discussion of investment principles in perfect capital markets characterised by

certainty. According to Fisher™s Separation ˜eorem (1930), it is irrelevant whether a company™s future

earnings are paid as a dividend to match shareholders™ consumption preferences at particular points

in time. If a company decides to retain pro˚ts for reinvestment, shareholder wealth will not diminish,

providing that:

-Management™s

minimum

required return on a project ˚nanced by retention (the discount rate,

r) matches the shareholders™

desired

rate of return (the yield, K

e) that they can expect to earn

on alternative investments of comparable risk in the market place, i.e. their

opportunity

cost

of capital.

-In the interim, shareholders can always borrow at the market rate of interest to satisfy their

income requirements, leaving management to invest current unpaid dividends on their behalf

to ˚nance future investment, growth in earnings and future dividends.

From the late 1950s, Myron J. Gordon developed Fisher™s theory that dividends and retentions are

perfect substitutes

by analysing the impact of di˛erent dividend and reinvestment policies (and their

corresponding yields and returns) on the current share price for all-equity ˚rms using the mathematical

application of a

constant growth

formula.

What is now termed the

Gordon dividend-growth model

de˜nes the current

ex-div

price of a share by capitalising next

year™s dividend at the amount by which the shareholders™ desired rate of return exceeds the constant annual rate of

growth in dividends.

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37 The Role of Dividend Policy

Using Gordon™s original notation where K

e represents the equity capitalisation rate; E

1 equals next

year™s post-tax earnings; b is the proportion retained; (1-b) E

1 is next year™s dividend; r is the return on

reinvestment and r multiplied by b equals the constant annual growth in dividends:

16. P0 = (1-b) E1 / Ke Œ rb subject to the proviso that K

e > rb for share price to be ˚nite.

Today, the equation™s notation is simpli˚ed in many Finance texts as follows, with D

1 and g representing

the dividend term and growth rate respectively, subject to the constraint that K

e > g.

17. P0 = D1 / Ke Œ gIn a

certain world

, Gordon con˚rms Fisher™s relationship between corporate reinvestment returns (r)

and the shareholders™ opportunity cost of capital (K

e). Share price only responds to pro˚table investment

(business) opportunities and not changes in dividend (˚nancial) policy because investors can always

borrow to satisfy their income requirements. To summarise the dynamics:

Shareholder wealth (price) will stay the same if r equals K

eShareholder wealth (price) will increase if r is greater than K

eShareholder wealth (price) will decrease if r is lower than K

eDownload free eBooks at bookboon.com

38 The Role of Dividend Policy

Activity 1

To con˜rm the impact of retention-˜nanced investment on share price de˜ned by Gordon under conditions of

certainty

, use the following data for Jovi plc with a full dividend distribution policy to establish its current share price.

EPS 10 pence

Dividend Yield 2.5%

Now recalculate price, with the same EPS forecast of 10 pence, assuming that Jovi revises its distribution policy. The

company reinvests 50 percent of earnings in projects with rates of return that equal its current dividend yield. Also

comment on your ˜ndings.

Ł Full Distribution (Zero Growth)

Without future injections of outside ˚nance, a forecast EPS of 10 pence and a policy of

full

distribution

(dividend per share also equals 10 pence) Jovi currently has a

zero growth rate

. Shareholders are satis˚ed with a 2.5 per cent yield on their investment. We can therefore

de˚ne the current share price, using either

constant

dividend or earnings valuations for

the capitalisation of a

perpetual annuity

, rather than a growth model, because they are all

˚nancially

equivalent

.P0 = E1 / Ke = D1 / Ke = 10 pence / 0.025 = D

1 / Ke Œ g = 10 pence / 0.025 Œ 0 = £4.00

Ł Partial Distribution (Growth)

Now we have the same EPS forecast of 10 pence but a reduced dividend per share. 50 percent of

earnings are reinvested in projects with rates of return equal to the current equity capitalisation

rate (yield) of 2.5 percent.

According to Gordon, dividends will

grow at a constant rate in

perpetuity

. ˜us, Jovi™s revised

current

ex-div

share price is determined by capitalising next year™s dividend at the amount by

which the desired rate of return exceeds the constant annual growth rate of dividends.

Using Equations (16) or (17):

P0 = (1-b) E1 / Ke Œ rb = P0 = D1 / Ke Œ g = 5 pence / 0.025 Œ 0.0125 = £4.00

Ł Commentary

Despite abandoning a constant share valuation in favour of the growth model to accommodate

a change in economic variables relating to dividends retention, reinvestment and growth, Jovi™s

share price remains the same.

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39 The Role of Dividend Policy

According to Gordon, this is because movements in share price relate to the pro˚tability of corporate

investment opportunities and not alterations to dividend policy. So, if the company™s rate of return on

reinvestment (r) equals the shareholders™ yield (K

e) price will not change. It therefore follows logically that:

Shareholder wealth (price) will only increase if r is greater than K

eShareholder wealth (price) will only decrease if r is lower than K

eActivity 2

Can you con˜rm the Gordon model™s prediction that if K

e = 2.5%, b = 0.5 but r moves from 2.5% to 4.0%, or down to

1.0%, then P0 moves from £4.00 to £10.00 or £2.50 respectively?

3.2 Gordon™s ‚Bird in the Hand™ Model

Gordon™s initial analysis of share price determination depends absolutely on the assumption of

certainty

. For example, our previous Activity data initially de˚ned a constant equity capitalisation rate (K

e) equivalent

to a managerial assessment of a constant return (r) on new projects ˚nanced by a constant

retention (b). ˜is ensured that wealth remained constant (e˛ectively Fisher™s Separation ˜eorem). We

then applied this mathematical logic to demonstrate that share price and hence shareholder wealth stays

the same, rises or falls only when:

Ke = r, K

e > r, K

e < rBut what if the future is

uncertain

?According to Gordon (1962) rational, risk averse investors should

prefer dividends earlier, rather than

later

(a ﬁbird in the handﬂ philosophy) even if retentions are more pro˚table than distributions (

i.e. r >

Ke). From period to period, they should also prefer

high dividends to low dividends.

˜us, shareholders

will discount near dividends and higher payouts at a lower rate, which is dated (K

et). In other words,

they require a higher overall

average

return on equity (K

e) from ˚rms that retain a higher proportion of

earnings, with obvious implications for share price. Expressed mathematically:

Ke = f ( Ke1 < Ke 2 < – Ke n )˜e equity capitalisation rate is no longer a

constant

but an

increasing

function of the

timing

and

size

of

a dividend payout. So, an

increased

retention ratio results in a

rise

in the discount rate (dividend yield)

and a

fall

in the value of ordinary shares:

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40 The Role of Dividend Policy

To summarise Gordon™s plausible

uncertainty

hypothesis, where dividend (˚nancial) policy, rather than

investment (business) policy, determines share price:

˙e lower the dividend, the higher the risk and the higher the yield, the lower the price

.Review Activity

According to Gordon, the theoretical policy prescription for an

all-equity

˜rm in a world of uncertainty is unambiguous.

Ł Maximise the dividend payout ratio and you minimise the equity capitalisation rate, which maximises share

price and hence shareholder wealth.

But from 1959 to 1963 Gordon published a body of theoretical and empirical work using real world stock market data

to prove his ﬁbird in the hand philosophyﬂ with con˚icting statistical results.

To understand why, analyse the two data sets below for Jovi plc in a world of

uncertainty

. The ˜rst represents a full

dividend policy distribution. The second re˚ects a rational managerial decision to retain funds, since the company™s

return on investment exceeds the shareholders™ increased capitalisation rate (Fisher™s theorem again).

1. Explain why the basic requirements of the Gordon growth model under conditions of uncertainty are satis˜ed.

2. Con˜rm whether the corresponding share prices are positively related to the dividend payout ratio, as

Gordon predicts.

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41 The Role of Dividend Policy

Dividend Policy, Growth and Uncertainty

Forecast EPS

Retention

Rate

Dividend Payout

Return on Investment

Growth Rate

Overall Shareholder

Returns E1(b)

(1-b)

(r)rb = g

Ke £0.1001.0--0.025 £0.100.50.50.0750.03750.050An Indicative Outline Solution

1. ˜e Basic Requirements

Under conditions of

certainty

Gordon asserts that movements in share price relate to the pro˚tability of

corporate investment and not dividend policy. However, in a world of

uncertainty

the equity capitalisation

rate is no longer a constant but an increasing function of the timing of dividend payments. Moreover,

an increase in the retention ratio results in a further rise in the periodic discount rate.

So far so good, since our data set satis˚es these requirements. Moving from full distribution to partial

distribution elicits a rise in K

e even though withholding dividends to ˚nance investment accords with

Fisher™s wealth maximisation criterion (r > K

e) and also satis˚es the mathematical constraint of the

Gordon growth model (K

e > rb).2. Has Share Price Fallen with Dividend Payout?

Rational, risk averse investors may prefer their returns in the form of dividends now, rather than later (a

ﬁbird in the handﬂ philosophy that values them more highly). But using the two data sets, which satisfy

all the requirements of the Gordon model under conditions of uncertainty, reveals that despite a change

in dividend policy, share price remains unchanged!

Uncertainty,

Di˜erential

Dividend and Growth Rates with a

Uniform Price: P

0 = (D1/Ke-g) = £4.00Forecast EPS

Retention

Rate

Dividend Payout

Return on Investment

Growth Rate

Overall Shareholder

Returns E1(b)

(1-b)

(r)rb = g

Ke £0.1001.0--0.025 £0.100.50.50.0750.03750.050Download free eBooks at bookboon.com

42 The Role of Dividend Policy

3.3 Summary and Conclusions

˜e series of variables in the previous table were deliberately chosen to ensure that share price remained

unchanged. But the important point is that they all satisfy the requirements of Gordon™s model, yet

contradict his prediction that share price should fall.

Moreover, it would be just as easy to provide another data set that satis˚es these requirements but produces

a rise in share price. No wonder Gordon and subsequent empirical researchers have o˝en been unable

to prove with statistical signi˚cance that

real world equity values

are:

Positively related to the dividend payout ratio

Inversely related to the retention rate

Inversely related to the dividend growth rate

Explained simply, Gordon confuses dividend policy (

˝nancial risk

) with investment policy (

business

risk

). For example, an increase in the dividend payout ratio, without any additional ˚nance, reduces a

˚rm™s operating capability and

vice versa

. Using Equation (17)

P0 = D1 / Ke Œ g˜e weakness of Gordon™s hypothesis is obvious. Change D

1, then you change K

e and g. So, how do

investors unscramble their di˛erential e˛ects on price (P

0) when all the variables on the

right hand side

of the equation are now a˛ected? And in our example, cancel each other out!

For the moment, suˆce it to say that Gordon encountered a very real world problem when testing his

theoretical model empirically. What statisticians term

multicolinearity

. Fortunately, as we shall discover,

two other academic researchers (Modigliani and Miller) were able to provide the investment community

with a more plausible explanation of the determinants of share price behaviour.

3.4 Selected References

1. Fisher, I.,

˙e ˙eory of Interest

, Macmillan (New York), 1930.

2. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

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43 MM and Dividends

4

MM and DividendsIntroduction

Under conditions of

certainty

, the Gordon growth model (P

0 = D1/Ke Œ g) reveals why share price

movements relate to the nature of a company™s pro˚table investment opportunities (business risk) and

not variations in its dividend policy (˚nancial risk). In a world of

uncertainty

, Gordon (1962) then

explains how price becomes a function of dividends. Rational, risk-averse investors prefer their returns

in the form of dividends now, rather than later (a ﬁbird in the handﬂ philosophy).

˜e purpose of this Chapter is to evaluate an alternative hypothesis developed by the joint Nobel Prize

winning economists, Franco Modigliani and Merton H. Miller (MM henceforth). Since 1961, their views

on the

irrelevance of dividend policy

when valuing shares based on the economic ﬁlaw of one priceﬂ have

dominated the subsequent development of modern ˚nance.

4.1 The MM Dividend Hypothesis

MM criticise the Gordon growth model under conditions of uncertainty supported by a wealth of

subsequent empiricism, notably the consultancy work of Joel M. Stern and G. Bennett Stewart 111

(Stern-Stewart) referenced by the author in Chapters Eight of

Strategic Financial Management

(2008) and its

Exercise

companion (2009). According to MM and their proponents, dividend policy is not a

determinant of share price in reasonably eˆcient markets because dividends and retentions are

perfect

economic substitutes

.If

shareholders

forego a current dividend to bene˜t from a future retention-˜nanced capital gain, they can still create

their own

home made dividends to match their consumption preferences by the sale of shares or personal borrowing

and be no worse o˛.

If a

company

chooses to make a dividend distribution, it too, can still meet its investment requirements by a new issue

of equity, rather than retained earnings. So, the e˛ect on shareholders™ wealth is also neutral.

Consequently,

business risk, rather than

˜nancial risk, de˜nes all investors and management need to know about corporate

economic performance.

˜eoretically and mathematically, MM have no problem with Gordon under conditions of

certainty

. ˜eir

equity capitalisation rate (K

e) conforms to the company™s class of business risk. So, as Fisher predicted

(1930) share price is a function of variations in pro˚table corporate investment and not dividend policy.

But where MM depart company from Gordon is under conditions of

uncertainty

.Download free eBooks at bookboon.com

44 MM and Dividends

As we concluded in Chapter ˜ree, Gordon confuses dividend policy with investment policy. For example,

an increase in the dividend payout ratio, without any additional ˚nance, reduces a ˚rm™s operating

capability and

vice versa

. MM also assert that because uncertainty is

non-quanti˝able

, it is logically

impossible to capitalise a

multi-period

future stream of dividends, where K

e1 < Ke2 < Ke3 –etc.

according

to the investors™ perception of the unknown, as Gordon recommends.

MM therefore de˚ne a current

ex-div

share price using the following

one period

model, where K

e equals

the shareholders™ desired rate of return (capitalisation rate) relative to the ﬁqualityﬂ of a company™s

periodic earnings (class of business risk). ˜e greater their variability, the higher the risk, the higher

Ke, the lower the price and

vice versa

.(18) P0 = D1 + P1 / 1 + Ke MM then proceed to prove that for a

given

investment policy of

equivalent

business risk (where K

e remains

constant) a change in dividend policy cannot alter current share price (P

0) because:

˙e next ex-div price (P

1) increases by a corresponding reduction in (D

1) and vice versa.

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45 MM and Dividends

Activity 1

To illustrate MM™s dividend irrelevancy hypothesis, let us reinterpret the stock exchange data for Jovi plc, initially applied

to Gordon™s growth model in Chapter Three.

-With an EPS of 10 pence a full dividend distribution policy and yield of 2.5 per cent, establish Jovi™s current

ex-div share price using Equation (18).

- Now recalculate this price, with the same EPS forecast of 10 pence, assuming that Jovi revises its dividend policy

to reinvest 100 percent of earnings in future projects with rates of return that equal its current yield.

With a policy of

full

dividend distribution, MM would de˚ne:

(18) P0 = D1 + P1 / 1 + Ke = £0.10 + £4.00 / 1.025 = £4.00Refer back to Chapter ˜ree and you will discover that this

ex-div

price is

identica

l to that established

using the Gordon growth model.

Turning to a policy of

nil

distribution (

maximum

retention) where pro˚ts are reinvested in projects of

equivalent business risk (

i.e. 2.5 per cent):

(18) P0 = D1 + P1 / 1 + Ke = £0 + £4.10 / 1.025 = £4.00According to MM, because the managerial cut-o˛ rate for investment still equals K

e, the

ex-div

price rise

matches the fall in dividend exactly, leaving P

0 unchanged.

You might care to con˚rm that using the Gordon growth model from the previous Chapter:

(17) P0 = D1 / Ke Œ g = 0In other words, if a company does not pay a dividend, which is not unusual (particularly for high-tech,

growth ˚rms), it is impossible to determine a share price.

4.2 The MM Hypothesis and Shareholder Reaction

You will also recall from Chapter ˜ree that even if Gordon™s model is mathematically de˚nable

(Ke>g and D

1>0) he argues that a

fall

in dividends should produce a

rise

in the equity capitalisation rate,

causing share price to

fall

. However, MM reject this argument.

If a company™s reduction in dividends fails to match shareholders™ expectations, they can always create

home-made

dividends by selling part of their holdings (or borrowing) to satisfy their consumption preferences, without a˛ecting

their overall wealth.

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46 MM and Dividends

To understand MM™s proposition, let us develop the data from Activity 1 using Equation (18) assuming

that the number of shares currently owned by an individual shareholder is de˚ned by (n) to represent

their holding.

(19) nP0 = nD1 + nP1 / 1 + Ke Activity 2

Assume you own a number of shares (n = 10,000) in Jovi plc and expect an initial managerial policy of full dividend

distribution. From the previous Activity and Equation (19) it follows that your current stock of wealth is worth:

nP0 = nD1 + nP1 / 1 + Ke = £1,000 + £40,000 / 1.025 = £40,000Now assume that the ˜rm withholds all dividends for reinvestment. What would you do, if your income requirements

(consumption preferences) equal the dividend foregone (£1,000)?

According to MM, the

ex-div

price should increase by the reduction in dividends. So, your holding is

now valued as follows, with no overall change:

(19) nP0 = nD1 + nP1 / 1 + Ke = £0 + £41,000 / 1.025 = £40,000However, you still need to satisfy your income preference for £1,000 at time period one.

So, why not sell 250 shares for £41,000 / 10,000 at £4.10 each?

You now have £1,025, which means that you can take the income of £1,000 and reinvest the balance of

£25 on the market at your desired rate of return (K

e=2.5%). And remember you still have 9,750 shares

valued at £4.10.

To summarise your new stock of wealth:

Shareholding 9,750: Market value £39,975: Homemade Dividends £1,000: Cash £25

Have you lost out?

According to MM,

of course not

, since future income and value are unchanged:

£ nP1 = 9,750 x £4.10 39,975 Cash reinvested at 2.5% 25 Total Investment 40,000 Total annual return at 2.5% 1,000 Download free eBooks at bookboon.com

47 MM and Dividends

MM conclude that if

shareholders do not like the heat they can get out of the kitchen

by selling an appropriate proportion

of their holdings, borrowing (or lending) to match their consumption (income) preferences.

4.3 The MM Hypothesis: A Corporate Perspective

Let us move from the shareholder to the company and what is regarded as the

proof

of MM™s dividend

irrelevancy hypothesis.

Usually, it is li˝ed

verbatim

from the mathematics of their original article (1961)

and relegated to an Appendix in the appropriate chapter of many ˚nancial texts, with little, if any,

numerical exposition. As I mentioned in Chapter One, this explains why ˚nance students are o˝en le˝

confused when revising for examinations and soon forget MM when they enter the real world of work.

To remedy this situation, let us it examine and apply the equations of MM™s proof in detail (using their

notation) with reference to the previous data for Jovi plc.

According to MM, dividends and retentions are

perfect economic substitutes,

leaving shareholder wealth una˛ected by

changes in distribution policy

. For its part too, a ˜rm can resort to new issues of equity to ˜nance any shortfall in its

investment plans without compromising its current

ex-div

price.

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48 MM and Dividends

To illustrate MM™s

corporate

proposition, assume a ˚rm™s total number of shares currently in issue equals

(n). We can de˚ne its

total market capitalisation of equity

as follows:

(19) nP0 = nD1 + nP1 / 1 + Ke Now assume the ˚rm decides

to distribute all earnings as dividends.

If investment projects are still to

be implemented, the company must therefore raise new equity capital equivalent to the proportion of

investment that is no longer funded by retentions.

According to MM, the number of new shares (m) issued at an ex

-div

price (P

1) must therefore equal the

total dividend per share retained (nD

1) de˚ned by:

(20) mP1 = nD1 Based on all shares

outstanding

at time period one (nP

1+mP1) MM then rewrite Equation (19) to represent

the total market value of

original

shares in issue as follows:

(21) nP0 = 1/ Ke [ nD1 + (n + m) P1 Œ mP1]And because mP

1 = nD1 this equation simpli˚es to:

(22) nP0 = 1/ Ke (n + m) P1MM therefore conclude that because the dividend term disappears from their market capitalisation of equity, it is

impossible to assert that share price is a function of dividend policy.

To illustrate the

corporate

dynamics of MM™s argument, let us develop the data from Activity 2, using

the preceding equations where the company™s total number of shares in issue equals (n).

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49 MM and Dividends

Activity 3

Assume Jovi plc begins the period with a

maximum

retention policy (

nil distribution) and a given investment policy.

Shares are therefore valued currently at £4.00 with an

ex-div

price of £4.10 at time period one as follows:

(18) P0 = D1 + P1 / 1 + Ke = £0 + £4.10 / 1.025 = £4.00If Jovi has one million shares in issue, we can also derive the company™s

total market capitalisation of equity

: (19) nP0 = nD1 + nP1 / 1 + Ke = £0 + £4.1m / 1.025 = £4m But now assume that the ˜rm decides

to distribute all earnings as dividends

(10 pence per share on one million issued)

without compromising investment (

i.e.

it is still a ﬁgivenﬂ).

Con˜rm that this policy leaves Jovi™s share price unchanged, just as MM hypothesise.

If investment projects are still to be implemented, the company must raise new equity capital equal to the

proportion of investment that is no longer funded by retained earnings. According to MM, the number

of new shares (m) issued

ex-div

at a price (P

1) must therefore equal the total dividend per share retained

(nD1) de˚ned by the following equation.

(20) mP1 = nD1 = £100,000Based on all shares

outstanding

at time period one (nP1+mP

1) we can rewrite Equation (19) representing

the total market value of

original

shares in issue as follows:

(21) nP0 = 1/ Ke [ nD1 + (n + m) P1 Œ mP1] ˜is simpli˚es to the following equation where

the dividend term disappears

.(22) nP0 = 1/ Ke (n + m) P1 = 1/ 1.025 (nP1 + £100,000) = £4 million

Since there is also only one unknown in the equation (P

1) then dividing through by the number of shares

originally in issue (n = one million) and rearranging terms, we revert to:

(18) P0 = D1 + P1 / 1 + Ke = P1 + £0.10 / 1.025 = £4.00And simplifying, solving for P

1:P1 = £4.00Download free eBooks at bookboon.com

50 MM and Dividends

˜us, as MM hypothesise:

-˜e

ex-div

share price at the end of the period has fallen from its initial value of £4.10 to

£4.00, which is exactly the same as the 10 pence rise in dividend per share, therefore leaving

P0 unchanged.

-Because the dividend term has disappeared from the equations, it is impossible to conclude

that share price is a function of dividend policy.

Review Activity

To rea˝rm the logic of the MM dividend irrelevancy hypothesis, revise the Jovi data set for a

nil distribution to assess

the implications for both the shareholders and the company if management now adopt a policy of

partial

dividend distribution, say 50 per cent?

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51 MM and Dividends

4.4 Summary and Conclusions

MM criticise the Gordon growth model under conditions of uncertainty from both a

proprietary

(shareholder) and

entity

(corporate)

perspective. ˜e current value of a ˚rm™s equity is

independent

of its dividend distribution policy, or alternatively its retention policy, because they are

perfect

economic

substitutes:

-˜e

quality

of earnings (business risk), rather than how they are

packaged

for distribution

(˚nancial risk), determines the shareholders™ desired rate of return and management™s cut-o˛

rate for investment (project discount rate) and hence its share price.

-If a company

chooses

to make a dividend distribution it can always meet its investment

requirements by a new equity, issue rather than use retained earnings, so that the e˛ect on

shareholders™ wealth is neutral.

-As a corollary, dividend policy should therefore be regarded as a

passive residual

, whereby

management return unused funds to shareholders because their search for new investment

opportunities cannot maintain shareholder wealth, subsequently con˚rmed by the

agency

principle of Jensen and Meckling (1976).

It therefore seems reasonable to conclude Part Two with the following practical observations on our analyses

of share valuation theories and their application by stock market participants, including management.

The P/E ratio (reciprocal of the earnings yield) associated with the pro˜tability of investment (

business risk

), rather than a

dividend yield associated with the periodic distribution of earnings (

˜nancial risk) published in the ˜nancial press and on

the internet, should encapsulate all the investment community needs to know about corporate economic performance.

4.5 Selected References

1. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

2. Miller, M.H. and Modigliani, F., ﬁDividend policy, growth and the valuation of sharesﬂ,

˙e

Journal of Business of the University of Chicago

, Vol. XXXIV, No. 4 October 1961.

3. Hill, R.A.,

Strategic Financial Management,

Chapter Eight,

bookboon.com

(2008).4. Hill, R.A.,

Strategic Financial Management: Exercises,

Chapter Eight,

bookboon.com

(2009).5. Fisher, I.,

˙e ˙eory of Interest

, Macmillan (New York), 1930.

6. Jensen, M.C. and Meckling, W.H., ﬁ˜eory of the Firm: Managerial Behaviour, Agency Costs

and Ownership Structureﬂ,

Journal of Financial Economics

, 3, October 1976.

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52 Part Three:

The Finance Decision

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53 Debt Valuation and the Cost of Capital

5

Debt Valuation and the

Cost of Capital

Introduction

Part Two

detailed the MM-Gordon controversy as to whether earnings or dividends are a prime

determinant of share value

Ł According MM™s ﬁlaw of one priceﬂ, the current value of an all-equity ˚rm is

dependent

upon

its investment strategy and

independent

of its dividend policy.

Ł ˜e variability of earnings (

business

risk), rather than how they are packaged for distribution

(˝nancial

risk), determines the shareholders™ desired rate of return (cost of equity) and

management™s cut-o˛ rate for investment (project discount rate) and hence its share price.

Part ˜ree

(the

Finance Decision

) now introduces MM™s earlier, consistent theory of capital structure

(1958) by relaxing our

all-equity

assumption to introduce an element of cheaper borrowing (debt) into

the corporate ˚nancial mix, premised on managerial policies designed to maximise shareholder wealth.

By reformulating the share valuation models of Part Two and introducing the pricing and return of

loan stock and other sources of ˚nance, a managerial cut-o˛ rate for project appraisal using an overall

weighted average cost of capital (WACC) will be derived. Given its assumptions and limitations, we shall

then consider the vexed question as to whether capital gearing (leverage) is a determinant of WACC and

total corporate value (the ﬁtraditionalﬂ view) or an irrelevance as MM hypothesise.

Based on their arbitrage concept, we shall arrive at two conclusions, which conform to MM™s dividend

irrelevancy position.

Ł Total corporate value (debt plus equity) represented by the expected NPV of a ˚rm™s income

stream discounted at a rate appropriate to its business risk, should be una˛ected by ˚nancial

risk associated with its mode of ˚nancing.

Ł Any rational debt-equity ratio should produce the same overall cut-o˛ rate for investment

(WACC) equivalent to the cost of equity in an all-equity ˚rm.

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54 Debt Valuation and the Cost of Capital

5.1 Capital Costs and Gearing (Leverage): An Overview

Firms rarely ˚nance capital projects by equity alone. ˜ey utilise long and short term funds from a

variety of sources at a variety of costs. No one source is free. As the following table reveals, some have

an explicit

cost but others have only an

implicit

or opportunity

cost. For example, loan issues are sourced

at an explicit market rate of interest, whereas the

marginal

cost of earnings retained for new investment

is measured by the current return

foregone

by shareholders.

Source of Finance

Capital Cost

Share Issues: Ordinary

Preference

Earning per share (EPS) or Dividends plus growth

Fixed Dividend

Loan Issues: Secured and Unsecured

Convertible

Interest payable plus any premium payable on repayment.

Present interest, plus future EPS (with normal conversion price typically

above current market price)

Retained earnings

Shareholder return: EPS or Dividends plus growth

Depreciation

Opportunity cost

Short-term borrowings

Market rate of interest

Deferred taxation

Opportunity cost

Deferred payments to creditors

Opportunity cost, plus any loss of goodwill and administrative costs

Reduction in stocks

Opportunity cost, plus any loss of goodwill and loss of sales

Reduction in debtors

As above

Debt factoring

Above base rate

Sale of excess or idle assets

Alternative yield

Sale of property and lease back

Leasing cost plus, any capital appreciation

Research and Development

Opportunity cost

Unallocated Overheads

Opportunity cost

Figure 5.1:

Sources of Finance and Capital Costs

Explicit or not, management must ˚rst identify the current (marginal) costs of each type of capital

employed (debt, as well as equity) to establish an

overall

cost of capital as a project discount rate for

corporate investment. ˜e component costs must then combine to derive a marginal,

weighted average

cost of capital

(WACC).

To simplify the conceptual computation of WACC (considered in Chapter Six) we shall restrict our

analyse to the impact of the value and cost (return) of the most signi˚cant alternative to equity as an

external source of ˚nance, namely corporate borrowing in the form of debentures (or corporate bonds

and loan stock to use American parlance).

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55 Debt Valuation and the Cost of Capital

Like other sources of debt and creditor ˚nance itemised in Table 5.1, corporate borrowing is attractive

to global management.

Ł Interest rates are typically lower than the cost of equity.

Ł Debt-holders accept lower returns than shareholders, because their investment is less risky.

Ł Unlike dividends, interest is guaranteed and a prior claim on pro˚ts.

Ł Debt-holders are also paid before shareholders from the sale of assets in the event of liquidation.

Ł In many countries, interest payments on debt also qualify for corporate tax relief, which does

not apply to dividends, thereby reducing their ﬁrealﬂ cost to the ˚rm.

Combine these factors and we can summarise the

traditional

approach to the corporate

˝nancing

decision,

which runs counter to MM™s theory of capital cost, ˚nance and investment outlined earlier.

According to traditionalists, the introduction of borrowing into a ˜rm™s ˜nancial structure, termed capital

gearing or leverage

, should lower the overall return (cut-o˛ rate) that management need to earn on new investments relative to

all-equity

funding. Consequently, the expected NPV of geared projects should rise with a fall in their discount rates,

producing a corresponding increase corporate wealth.

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56 Debt Valuation and the Cost of Capital

5.2 The Value of Debt Capital and Capital Cost

As marketable securities, the principles of loan valuation are similar to those for equity but less

problematical. Stock is issued above, below or at

par

value depending on economic conditions. However,

the annual cash return is known from the outset. It always equals a speci˚c rate of interest relative to

par value (termed the

coupon rate

or

nominal

yield). ˜e stock™s life might also be speci˚ed in advance

with a guaranteed capital repayment (i.e.

redeemable

as opposed to

irredeemable

debt). Ignoring tax for

the moment:

Ł ˜e current price of any debenture (bond) is determined by a summation of future interest

payments, plus the redemption price (if applicable) all discounted back to a present

value (PV).

Ł ˜e annual cost of corporate debt or yield (to redemption if applicable) is the discount rate

that equates current price to these expected future cash ˙ows, namely their Internal Rate of

Return (IRR) using discounted cash ˙ow (DCF) analysis.

In the case of irredeemable debentures, about to be issued or subsequently trading at par, the market

price and IRR obviously equal the par value and

coupon rate

respectively. However, if

price di˜ers from

par value

, either at issue or when the debt is later traded, the

IRR no longer equals the coupon rate

. To

see why, let us de˚ne the price of debt (P

0) at any point in time.

(1) P0 = I / 1+Kd + I / (1+Kd)2 + – I / (1+Kd)Where: I = interest (the coupon rate expressed in money terms) received per annum in perpetuity

Kd = the company™s annual cost of debt de˚ned as an IRR percentage.

Since the annual interest payment is ˚xed in perpetuity, Equation (1) simpli˚es to the familiar valuation

formula for a level annuity: interest divided by current market price:

(2) P0 = I / KdIf we rearrange terms, the cost of debt equals the investment™s IRR de˚ned as the annual money interest

divided by current market price:

(3) Kd = I / P0And because interest (I) is constant year on year, it follows that if P

o rises (or falls) then K

d must fall (or

rise) proportionately.

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57 Debt Valuation and the Cost of Capital

Turning to redeemable stock, the nominal return to debt-holders in the year of redemption will be upli˝ed

by the redemption price payable. ˜us, when debt is issued or whenever investors trade debentures, the

current yield (K

d) is found by solving for the IRR in the following

˝nite

equation.

(4) P0 = [(I / 1 + Kd) + (I / 1 + Kd)2 –.+–. (I + Pn / 1 + Kd)n] Rewritten as follows:

(5) The World of Modigliani and Miller

57 Debt Valuation and the Cost of Capital

Turning to redeemable stock, the nominal return to debt-holders in the year of redemption will be upli˜ed

by the redemption price payable. ˚us, when debt is issued or whenever investors trade debentures, the

current yield (K

d) is found by solving for the IRR in the following

˜nite

equation.

(4) P0 = [(I / 1 + Kd) + (I / 1 + Kd)2 –.+–. (I + Pn / 1 + K

d)n] Rewritten as follows:

(5) Where: n = the number of periods to redemption,

Pn = the redemption value at time period n.

Irrespective of whether debt is redeemable, irredeemable, currently traded or about to be issued:

-˚e cost of capital (K

d) always

equals an internal rate of return (IRR).

-˚e IRR equates current price to the discounted future cash receipts that the loan stock produces.

-Only if the current price and redemption value (if any) equal the par value, will the IRR equal

the coupon rate (nominal yield).

If a debt issue has a coupon rate which is below the prevailing market rate of interest de˛ned by its

current IRR then by de˛nition current market value (price) will be below par value and

vice versa

.Activity 1

Manipulate the previous equations to calculate current debt yields if a company issued:

- £100 irredeemable debentures with a 10 percent coupon rate

- £100 debentures with the same coupon rate, redeemable at par ten years hence

You may assume that in both cases, similar debentures currently trade below par at £90.00 (conventionally termed as

£90 per cent).

What do these calculations mean to investors and corporate management?

Given current market conditions both £100 issues must be priced at £90 to ensure full subscription.

If

irredeemable

debentures are issued at £90 percent with a

money

coupon rate of £10 per annum, it

follows from Equation (3) that the current yield or cost of debt:

Kd = £10 / £90 = 11.1% Pn / (1 + Kd)nWhere: n = the number of periods to redemption,

Pn = the redemption value at time period n.

Irrespective of whether debt is redeemable, irredeemable, currently traded or about to be issued:

-˜e cost of capital (K

d) always

equals an internal rate of return (IRR).

-˜e IRR equates current price to the discounted future cash receipts that the loan stock produces.

-Only if the current price and redemption value (if any) equal the par value, will the IRR equal

the coupon rate (nominal yield).

If a debt issue has a coupon rate which is below the prevailing market rate of interest de˚ned by its

current IRR then by de˚nition current market value (price) will be below par value and

vice versa

.Activity 1

Manipulate the previous equations to calculate current debt yields if a company issued:

- £100 irredeemable debentures with a 10 percent coupon rate

- £100 debentures with the same coupon rate, redeemable at par ten years hence

You may assume that in both cases, similar debentures currently trade below par at £90.00 (conventionally termed as

£90 per cent).

What do these calculations mean to investors and corporate management?

Given current market conditions both £100 issues must be priced at £90 to ensure full subscription.

If

irredeemable

debentures are issued at £90 percent with a

money

coupon rate of £10 per annum, it

follows from Equation (3) that the current yield or cost of debt:

Kd = £10 / £90 = 11.1%Download free eBooks at bookboon.com

58 Debt Valuation and the Cost of Capital

If

redeemable

ten year debt was issued at the same price with the same coupon rate, we must derive the

current yield by solving for IRR using Equation (5).

The World of Modigliani and Miller

58 Debt Valuation and the Cost of Capital

If

redeemable

ten year debt was issued at the same price with the same coupon rate, we must derive the

current yield by solving for IRR using Equation (5).

Now the annual cost of debentures K

d is approximately 11.8%.

For the investor

, both debenture formulae perform the same functions as the equity models presented in

Chapter Two. Even though interest is ˜xed and a redemption date may be speci˜ed, debentures can be

traded at either a premium or a discount throughout their life. ˚us, the current rate of interest, like an

equity yield, is only a guide to the

true

return on life-time investment. In a world of uncertainty it can

only be determined by incorporating the capital gain or loss

retrospectively

when the security is sold. In

the case of redeemable debentures, held from issue through to redemption, this

ex-post return

calculation

is termed the

yield to maturity

or

redemption yield

.˚e current yield on debentures K

d therefore represents the return from holding the investment, rather

than selling at its current market price. It is an implicit

opportunity cost of capital

, because it is the

minimum return below which debenture holders could transfer their funds elsewhere for a market rate

of interest of equivalent risk, (Fisher™s Separation ˚eorem,

op. cit

.).Now the annual cost of debentures K

d is approximately 11.8%.

For the investor

, both debenture formulae perform the same functions as the equity models presented in

Chapter Two. Even though interest is ˚xed and a redemption date may be speci˚ed, debentures can be

traded at either a premium or a discount throughout their life. ˜us, the current rate of interest, like an

equity yield, is only a guide to the

true

return on life-time investment. In a world of uncertainty it can

only be determined by incorporating the capital gain or loss

retrospectively

when the security is sold. In

the case of redeemable debentures, held from issue through to redemption, this

ex-post return

calculation

is termed the

yield to maturity

or

redemption yield

.˜e current yield on debentures K

d therefore represents the return from holding the investment, rather

than selling at its current market price. It is an implicit

opportunity cost of capital

, because it is the

minimum return below which debenture holders could transfer their funds elsewhere for a market rate

of interest of equivalent risk, (Fisher™s Separation ˜eorem,

op. cit

.).Download free eBooks at bookboon.com

59 Debt Valuation and the Cost of Capital

For the company

, a successful debenture

issue must therefore match the risk-return pro˚le (yield) of loan

stock currently trading on the market. In an untaxed economy (more of which later) this rate of interest

required by investors represents the company™s

marginal

cost of capital for this fund source. As such, K

d is the relevant measure for assessing any new project ˚nanced by loan stock.

Returning to our previous Activity

, if management wish to maximise corporate wealth using expected

(ENPV) criteria then the 10 per cent coupon rate (nominal yield) is irrelevant. To be more precise, new

projects should be ˚nanced by irredeemable debt at a ﬁrealﬂ cost of 11.1 per cent discount rate, rather

than redeemable debt with a cost of 11.8 per cent.

Ł ˜e lower the discount rate, the higher the ENPV and vice versa. So at one extreme, a project

discounted at the coupon rate might be accepted, whilst at the other, the redeemable rate signals

rejection. Either way, corporate wealth is compromised; with a worst case scenario where the

cash ˙ows for a project™s acceptance using the coupon rate as a discount rate will not service

debt, forcing the ˚rm into liquidation.

To conclude, projects ˚nanced by debt (just like equity) should always be evaluated using a

marginal

cost of capital and not the

nominal

yield Only if the incremental return equals the current yield will the

marginal cost of raising additional ˚nance equal the current cost of capital in issue and attract investors.

5.3 The Tax-Deductibility of Debt

Whilst tax regimes di˛er throughout the world, one policy many governments have in common that

we need to consider is the treatment of debenture interest as an allowable deduction against a ˚rm™s tax

liability. As mentioned earlier, not only does this lower the ﬁtrueﬂ cost of corporate borrowing but also

widens the gap between yields on debt and equity.

Providing management can generate su˝cient taxable pro˜ts to claim the tax relief on debt interest, the higher the rate

of corporation tax, the greater the ˜scal bene˜t conferred on the company through issuing debt, rather than equity to

˜nance its investments.

In the preceding valuation models K

d represents the

gross

return received by investors

before

satisfying

their

personal

tax liability. What is important to the company, however, is the project discount rate de˚ned

by this gross return

aˆer corporation tax

.To prove the point, let us ˚rst consider

irredeemable

debt (i.e. with no redemption value) with a level

interest stream in perpetuity. ˜e valuation model

incorporating

tax is given by:

(6) P0 = I (1-t) / Kd t Download free eBooks at bookboon.com

60 Debt Valuation and the Cost of Capital

Where:

P0 = the current market price of debt,

I = annual interest payments

t = rate of corporation tax

Kd t = post-tax cost of debt

So, if we rearrange terms, the ﬁrealﬂ cost of debt to the company a˝er tax is

(7) Kd t = I (1-t) / P0And because the investors™

gross

return (K

d) equals the company™s cost of debt before tax, it follows that

with a tax rate (t) we can also rewrite Equation (7) as follows;

(8) Kd t = Kd (1-t)In a world of corporate taxation, the capital budgeting implications for management are clear.

(9) Kd t < Kd To maximise corporate wealth, the post-tax cost of debt should be incorporated into any overall discount

rate as a cut-o˛ rate for investment.

Equation (6) onwards might seem strange, since P

0 is still the market value of the debentures held by

investors represented by the future cash ˙ows which they expect to receive. But it is important to remember

that we are now modelling income-value relationships from the

company™s

perspective.

˜e interest cash ˙ows capitalised on the right-hand side of Equation (6) are therefore

net of corporation

tax, which do not concern investors directly. So, if a company pays £100,000 a year interest on irredeemable

debentures with a market price of £1 million and the rate of corporation tax is 25 percent, its e˛ective

cost of debt de˚ned by Equation (7):

Kd t = [£100,000 (1-0.25)] / £1 million = 7.5%

Turning to

redeemable

debt, the company still receives tax relief on interest. But o˝en the redemption

payment is not allowable for tax. To calculate the post-tax cost of capital, it is necessary to derive an IRR

that incorporates tax relief on interest alone by solving for K

d t in the following

˝nite

equation:

The World of Modigliani and Miller

60 Debt Valuation and the Cost of Capital

Where:

P0 = the current market price of debt,

I = annual interest payments

t = rate of corporation tax

Kd t = post-tax cost of debt

So, if we rearrange terms, the ﬁrealﬂ cost of debt to the company a˜er tax is

(7) Kd t = I (1-t) / P0And because the investors™

gross

return (K

d) equals the company™s cost of debt before tax, it follows that

with a tax rate (t) we can also rewrite Equation (7) as follows;

(8) Kd t = Kd (1-t)In a world of corporate taxation, the capital budgeting implications for management are clear.

(9) Kd t < Kd To maximise corporate wealth, the post-tax cost of debt should be incorporated into any overall discount

rate as a cut-o˚ rate for investment.

Equation (6) onwards might seem strange, since P

0 is still the market value of the debentures held by

investors represented by the future cash ˛ows which they expect to receive. But it is important to remember

that we are now modelling income-value relationships from the

company™s

perspective.

˝e interest cash ˛ows capitalised on the right-hand side of Equation (6) are therefore

net of corporation

tax, which do not concern investors directly. So, if a company pays £100,000 a year interest on irredeemable

debentures with a market price of £1 million and the rate of corporation tax is 25 percent, its e˚ective

cost of debt de˙ned by Equation (7):

Kd t = [£100,000 (1-0.25)] / £1 million = 7.5%

Turning to

redeemable

debt, the company still receives tax relief on interest. But o˜en the redemption

payment is not allowable for tax. To calculate the post-tax cost of capital, it is necessary to derive an IRR

that incorporates tax relief on interest alone by solving for K

d t in the following

˜nite

equation:

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61 Debt Valuation and the Cost of Capital

Consider ˚ve-year debt with a 15 percent coupon rate, redeemable at £100 par, issued at £90 percent. If

the annual rate of corporation tax is 33 percent, we can determine the post-tax cost of debt by solving

for K

dt in the following equation.

The World of Modigliani and Miller

61 Debt Valuation and the Cost of Capital

Consider ˜ve-year debt with a 15 percent coupon rate, redeemable at £100 par, issued at £90 percent. If

the annual rate of corporation tax is 33 percent, we can determine the post-tax cost of debt by solving

for Kd t in the following equation.

Activity 2

A company™s irredeemable debt has a coupon rate of 8 percent and a market value of £76 percent.

Corporation tax is 30 percent and the ˜rm™s has su˚cient tax liability to set o˛ against its interest.

Calculate

the investor™s gross return and the company™s e˛ective cost of debt.

Comment

on the disparity between the two and the capital budgeting implications for management.

Investors receive the following gross return before personal taxation:

Kd = £8 / £76 = 10.53%Activity 2

A company™s irredeemable debt has a coupon rate of 8 percent and a market value of £76 percent.

Corporation tax is 30 percent and the ˜rm™s has su˝cient tax liability to set o˛ against its interest.

Calculate

the investor™s gross return and the company™s e˛ective cost of debt.

Comment

on the disparity between the two and the capital budgeting implications for management.

Investors receive the following gross return before personal taxation:

Kd = £8 / £76 = 10.53%Download free eBooks at bookboon.com

62 Debt Valuation and the Cost of Capital

˜e post-tax cost to the company for providing this return is:

Kd t = £8(1-0.30) / £76 = 7.36%Loan interest reduces the corporate tax bill. For every £8 distributed to investors as interest, the company

e˛ectively pays:

I (1-t) = £8 (1-0.30) = £5.60˜e £2.40 di˛erence represents tax relief contributed by the tax authorities.

Turning to capital budgeting, if management ˚nance new investment by issuing debt, this must re˙ect

current post-tax yields of equivalent risk. Each £100 block will be priced at £76. ˜e post-tax cost of

debt capital (K

dt

=7.36%) represents the discount rate that equates the amount raised to the PV of future

cash ˙ow required to service this new issue (interest less tax relief).

The tax adjusted cost of debt (K

dt) is the IRR that represents the true corporate cost of new debt issues. If the ENPV of a

prospective debt-˜nanced project discounted at this IRR is positive, then its return will exceed the cost of servicing that

debt and management should accept it.

5.4 The Impact of Issue Costs on Equity and Debt

˜e introduction of a tax bias into our analysis of the cost of debt is our ˚rst example of a barrier to

trade that runs counter to Irving Fisher™s world of perfect competition outlined in Chapter One. But in

the ﬁrealﬂ world there are others, one of which we must now consider, namely issue costs.

In Chapter Four we hypothesised that dividends and earnings are

perfect economic substitutes

. At

the beginning of this chapter we also stated that the cost of retained earnings is best measured by an

opportunity cost, namely the shareholders™ return foregone. But even if we ignore the MM dividend-

earnings debate, how do we measure this?

In imperfect markets, a fundamental di˛erence between a new issue of ordinary shares (like any other

˚nancial security) and retained earnings are the

issue costs

associated with the former. As a consequence,

the marginal cost of equity issues is more expensive than retentions, which explains why management

hold back earnings for reinvestment

To prove the point, using previous notation and our knowledge of equity valuation for a constant dividend

stream (D) in perpetuity, let us introduce issue costs (C) into the

constant dividend valuation model.

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63 Debt Valuation and the Cost of Capital

˜e

marginal

cost of an ordinary share P

o issued by a company is now given by:

(11) Ke = D / Po (1 Œ C)By de˚nition, this is higher than the cost of retained earnings, since the latter do not incur issue costs.

˜e cost of retained earnings is simply equivalent to the current dividend yield forgone by

existing

shareholders, namely their opportunity cost of capital:

(12) Ke = D / PoNote that also, if we substituted earnings (E) for dividends into both of the previous equations;

management™s preference for retentions, rather than dividend distributions, would still prevail in the

presence of transaction costs.

Returning to the cost of loan stock, issue costs also increase the marginal cost of capital. ˜is is best

understood if we ˚rst substitute issue costs (C) into the cost of

irredeemable

debt in a

taxless

world. Like

the equity model, the denominator of Equation (13) is reduced by issue costs.

(13) Kd = I / P0 (1-C

)If we now assume that debt interest is tax deductible, the post-tax cost of debt originally given by

Equation (7) also rises.

(14) Kd t = I (1-t) / P0 (1-C)Review Activity

In preparation for Chapter Six and the information required to derive a weighted average cost of capital (WACC) as a

marginal cut-o˛ rate for investment, use the data below for B.Ferry plc to calculate:

1. The

total

market value of the company™s equity plus debt.

2. The

marginal

cost of each fund source.

The Data Set

-5 million ordinary £1 shares (common stock) currently quoted at £1.20,

-£6 million in retained earnings,

-4 million preference shares currently quoted at 60 pence,

-£2 million debentures (loan stock) currently trading below par at £80,

-Ordinary and preference shares currently yielding 20 and 10 per cent, respectively,

-Ordinary dividend growth of 5 per cent per annum,

-New issues costs of 20 pence per share for ordinary and preference shares,

-A pre-tax debt yield of 10 per cent,

-A 20 per cent rate of corporation tax.

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64 Debt Valuation and the Cost of Capital

An Indicative Outline Solution

1. Total Market Value

˜e overall market value for B.Ferry plc equals the summation of ordinary shares, retained earnings,

preference shares and debentures. With the exception of retained earnings (£6m) which are derived from

historical

cost based accounts, all capital issues are valued at their

market

price as follows:

(5m × £1.20) + £6m + (4m × £0.60) + (£2m × 0.80) = £16m2. Marginal Component Costs

˜e capital cost of each fund source is based on

market

value, not

book

(nominal or par) value because

management require today™s yields to vet new projects. Component costs should therefore be underpinned

by current returns for each category of investor who may ˚nance projects.

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65 Debt Valuation and the Cost of Capital

However, Ferry™s ultimate concern, (rather than investors) is its own

break-even

income stream that may

di˛er from the multiplicity of views held by proprietors and creditors. Consequently, the ˚rm™s component

costs not only incorporate any tax e˛ects, but also the costs of capital issues as follows:

Issue of ordinary shares

= Dividend / Net proceeds of issue, plus the growth rate

= [(£0.24 / £1.00] + 5% = 29% Retained earnings

= Dividend yield, plus the growth rate

= 20% + 5% = 25% Preference share issue

= Dividend / Net proceeds of issue

= £0.06 / £0.40 = 15%Debentures (a˝er tax)

= (interest /net proceeds of issue) multiplied by (1-tax rate)

= (£10.00 / £80.00) × (1 Œ 0.20) = 10% 5.5 Summary and Conclusions

In Chapter One, our study of strategic ˚nancial management began with a hypothetical explanation of

a company™s overall cost of capital as an investment criterion, designed to maximise shareholder wealth.

By Chapter Four, we demonstrated that an

all equity

company should accept capital projects using the

marginal cost of equity as a discount rate, because the market value of ordinary shares will increase by

the project™s NPV.

In this Chapter we considered the implications for project discount rates if funds were obtained from a

variety of sources other than the equity market, each of which requires a rate of return that may be unique.

For the purpose of exposition, we analysed the most signi˚cant alternative to ordinary shares as an

external source of funding, namely redeemable and irredeemable loan stock. We observed that corporate

borrowing is attractive to management because interest rates on debt are typically lower than equity

yields. ˜e impact of corporate tax relief on debenture interest widens the gap further, although the

tax-deductibility of debt is partially o˛set by the costs of issuing new capital, which are common to all

˚nancial securities.

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66 Debt Valuation and the Cost of Capital

In this newly leveraged situation, the company™s overall cost of capital measured by a weighted average

cost of capital (WACC) would seem to be a more appropriate investment criterion, rather than its cost

of equity.

So, given the solution to your latest Review Activity, let us move on to Chapter Six and formally analyse

how management can combine the component capital costs from various fund sources to derive a WACC

as an overall discount rate for project appraisal. ˜erea˝er, we shall explain MM™s startling contributions

to the subject.

5.6 Selected References

1. Miller, M.H. and Modigliani, F., ﬁDividend policy, growth and the valuation of sharesﬂ,

˙e

Journal of Business of the University of Chicago

, Vol. XXXIV, No. 4 October 1961.

2. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

3. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

4. Fisher, I., ˜e ˜eory of Interest, Macmillan (New York), 1930.

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67 Capital Gearing and the Cost of Capital

6

Capital Gearing and the

Cost of Capital

Introduction

If an

all-equity

company undertakes a capital project using the marginal cost of equity as its discount

rate, the total market value of ordinary shares should increase by the project™s expected NPV (ENPV).

However, most ˚rms use a

mix

of ownership capital and borrowed funds for new investments. ˜e

relationship between the two is termed

capital gearing

or

leverage

. A company is highly geared (levered)

when it has a signi˚cant proportion of borrowing relative to shares in its capital structure. It is low geared

when the ratio of debt to equity is small.

In Chapter Five we observed that corporate borrowing is attractive to management because interest rates

on debt are typically lower than equity. ˜is arises because capital market providers perceive debt as a

less risky investment than equity. Interest is paid before shareholders™ dividends and creditors also have

a prior claim on a ˚rm™s assets in the event of liquidation. Moreover, interest on debt o˝en quali˚es for

corporate tax relief. As a consequence:

Ł A judicious amount of debt introduced into a ˚rm™s capital structure should lower the overall

weighted average cost of capital

(WACC) employed as a cut-o˛ rate for the appraisal of new

projects, thereby increasing their ENPV and total corporate value.

You will also recall from Chapter Five that a company™s component capital costs are derived by identifying

the

opportunity cost

of each fund source using an appropriate valuation model that determines debt and

equity yields. Given the normative assumption that management should maximise pro˚t at minimum

cost, we shall now extend our analysis to answer three questions.

-How do individual capital costs combine to de˜ne WACC for use in investment appraisal?

-How valid are the theoretical assumptions that underpin WACC computations?

-What are the real-world problems associated with WACC estimations?

6.1 The Weighted Average Cost of Capital (WACC)

Let us begin our analysis by ˚rst de˚ning an overall cost of capital in a

taxless

world where management

has access to only two sources of ˚nance: equity and debt.

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68 Capital Gearing and the Cost of Capital

A general formula for WACC is given by the formula for a

simple weighted average

: (15) K = Ke (VE / VE + VD) + Kd (VD / VE + VD)Where:

K = WACC,

Ke = cost of equity,

Kd = cost of debt,

VE = market value of equity,

VD = market value of debt.

If we now introduce corporate taxation (at a rate t) the a˝er tax cost of debt K

d t should be substituted

into the preceding equation using the appropriate debt formulae from Chapter Six as follows.

(16) K = Ke (VE / VE + VD) + Kd t (VD / VE + VD)˜is is equivalent to:

(17) K = Ke (VE / VE + VD) + Kd (1-t) [(VD / VE + VD )]Equations (16) and (17) may be rewritten using simpler notation. For example, with tax:

(18) K = Ke (WE) + Kdt (WD)Where: W

E = the weighting applied to equity (V

E / VE + VD) WD = the weighting applied to debt (V

D / VE + VD) ˜us, a ˚rm ˚nanced equally by equity and debt yielding 10 percent and 5 percent, respectively, would

calculate its WACC using Equation (18) as follows:

K = 10% (0.5) + 5% (0.5) = 7.5%Activity 1

Given the following company data:

Ke = 12%, Kd = 8%, V

E = £6 million, V

D = 4 millionCalculate the WACC and write down your thoughts on any assumptions that might validate its use as a discount rate

for project appraisal before reading the next section.

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69 Capital Gearing and the Cost of Capital

˜e individual costs of equity and debt capital are weighted by their proportion of the company™s total

market value. Using Equation (18) and simplifying:

K = [(0.12 × 0.6) + (0.08 × 0.4)] / 1.0 = 0.104˜us, the WACC applied by management as a discount rate for new project appraisal is 10.4 percent.

6.2 WACC Assumptions

˜e use of WACC as a corporate discount rate depends upon three fundamental assumptions.

Ł New projects must have the same homogenous risk-return pro˚le as existing activities.

Ł Each project is marginal to the scale of the ˚rm™s existing operations.

Ł ˜e company will retain its existing capital structure, leaving ˚nancial risk unchanged.

˜e reason for the ˚rst assumption is obvious. A company™s component capital costs re˙ect the variability

of future expected dividend and interest ˙ows. ˜us, it follows, that WACC must also re˙ect the

overall risk of these combined ˙ows. So, if we use this ˚gure as a discount rate in project appraisal, the

new investment™s risk-return characteristics must satisfy the company™s existing expected dividend and

interest payments.

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70 Capital Gearing and the Cost of Capital

˜e second assumption is also common sense. When ˚rms consider new investment, the relevant costs

refer to the returns that the company must earn on relatively small incremental additions to its total

capital base. From an economic viewpoint, they are

marginal

costs of capital and are only applicable to

the appraisal of marginal investments: projects that are small relative to the size of the company.

Finally, the third assumption is necessary because WACC can only provide an appropriate discount rate

if new projects are ˚nanced in the

same proportion

as existing assets. ˜is arises for two reasons.

If a company alters its capital structure, the weights applied to the component costs in the WACC calculation would also

change, leading to a new discount rate.

A change in the capital mix (gearing) might also a˛ect the investors™ perception of the

˜nancial risk associated with

their investment in the ˜rm. They may then react by buying or selling (as opposed to holding) their securities, thereby

a˛ecting the respective yields which determine the WACC.

For example, a new debt issue could increase the uncertainly experienced by the shareholders when

they recognise that debt-holders will receive their claim to earnings (interest) before any dividend

distribution. With increased risk, they sell their holding and equity prices may fall because the market

requires a higher return as compensation. For the ˚rm, what seems a simple change in the debt-equity

ratio is, therefore, a complex decision. Quite apart from revised weightings at new market prices, it must

also consider the explicit

marginal

cost of issuing debt

and

the

implicit

cost to the shareholders of their

increased ˚nancial risk. All three may combine to produce a drastic change in WACC.

Activity 2

Changes in the ˜nancial mix (gearing) of a company and the impact of risk on its overall cost of capital and value do

not necessarily invalidate the use of WACC as an investment criterion.

Can you think of any reasons for this?

Whilst corporate investment decisions should determine a ˚rm™s overall cost of capital, management

should avoid the mistake of always associating the explicit marginal costs of new capital issues with a

speci˚c project. O˝en it will be diˆcult, if not impossible to assign a particular project to a particular

source of ˚nance. A company™s funds should therefore be viewed

collectively

. In as much as ˚nance is withdrawn from a

pool

of funds to invest in new projects, the pool is replenished

as fresh capital is raised from outside, or pro˚ts are retained. ˜us, the cost of capital used for any

particular project is not the cost of a

speci˝c

source of funds, but the

overall

cost of the company™s pool:

namely its WACC.

In the short run, it is frequently the case that certain funds might also be secured at advantageous rates

depending upon prevailing market conditions. ˜is will encourage ˚rms to depart brie˙y from their

long-run capital structure. Under such circumstances, however, WACC still represents an appropriate

discount rate for long-term investment, providing the projects exhibit a similar risk-return pro˚le.

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71 Capital Gearing and the Cost of Capital

Even if funds are raised explicitly from one source to ˚nance an incremental investment, there are

sound reasons for using the WACC as a discount rate, particularly if the change in the capital structure

represents a short-run deviation from the desired capital mix.

First, a rational choice of funds is a

˝nancial

decision taken in relation to the ˚rm™s long-term capital

structure and not the

investment

decision. Second, there are substantial economies of scale to be gained

in terms of reduced issue costs by raising large amounts of capital from one source and then another.

6.3 The Real-World Problems of WACC Estimation

Given the assumptions of

homogenous

risk,

marginal

investment and a

stable

capital structure, WACC

seems an appropriate

minimum

return criterion for new projects that will hopefully

maximise

wealth.

However, a company™s overall cost of capital is a complex concept, which may include far more than

shareholder dividend-growth expectations and ˚xed rates of debt interest. Moreover, the WACC model

assumes that once they are determined, the variables selected for inclusion in the model are correctly

de˚ned and will not change. But think about it?

WACC applied to investment projects extends over numerous time periods. ˜us, its value is likely to

change with economic circumstances, thereby invalidating original NPV calculations. A simple problem

concerns the estimation of a˝er-tax loan costs determined by an existing tax regime that government

revises. More complex, is stock market volatility. ˜e 2008 global ˚nancial meltdown with falling interest

rates, equity yields and security values was characterised by the market™s aversion to ˚nancing even ﬁblue

chipﬂ ˚rms. Yet by 2015, equity markets had recovered to an all-time high.

Even if we ignore recent dramatic events, it is important to realise that at any point in

normal

economic

cycles, the cost of capital and ˚nancial mix for individual companies can vary considerably, even within

the same sector. Some ˚rms are naturally more risky than others. Di˛erent companies may have di˛erent

capital structures, by accident if not design. As we shall discover, di˛erences in WACC have important

consequences for the relative economic performance of companies and wealth creation.

Review Activity

You are asked to evaluate a marginal investment costing £100,000 and yielding £11,500 per annum for the foreseeable

future, subject to the constraints that its acceptance will not alter the ˜rm™s existing risk-return pro˜le and capital structure:

1. Derive and explain WACC as a discount rate if the corporate tax rate is 25 percent.

2. Evaluate the project™s viability by applying the NPV decision rule.

3. Outline the implications for shareholder wealth maximisation.

The following information is available:

(i) Existing Capital Structure (£000 at cost)

Ordinary shares (12 million)

12,000Retained earnings

4,0006% Preference shares

2,0006% Irredeemable debt

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72 Capital Gearing and the Cost of Capital

Review Activity (cont)

(ii) Ordinary Shares

The current market price (ex div) is £7.00. Forecast total dividends are £6 million, which represent 75 percent of earnings.

Dividends have been growing at an annual compound rate of 5 percent. If new ordinary shares were issued now the

costs incurred would represent 25 pence per share and a reduction below market value of 50 pence per share would

also be required to ensure full subscription.

(iii) Preference Shares

Despite a par value of £1.00, current trades are only at 43 pence, with new issues at 40 pence.

(iv) Debentures

£100 loan stock currently priced at £92 would need to be issued at £90 percent

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73 Capital Gearing and the Cost of Capital

An Indicative Outline Solution

1. ˜e WACC Discount Rate

˜e derivation of WACC is straightforward. Using the appropriate capitalisation formulae, explained in

previous Chapters, including the Gordon growth model, tax and issue costs where appropriate:

Ł Marginal

component costs are de˚ned as follows:

Issue of ordinary shares

= (dividend per share / net proceeds of issue) + growth rate

= (0.50 / 6.25) + 0.05 = 13%Retained earnings

= dividend yield + growth rate

= (0.50 / 7.00) + 0.05 = 12.1%Preference Shares

= dividend per share / net proceeds of issue

= 0.06 / 0.40 = 15%Debentures (post-tax)

= [interest per debenture (1 Œ tax rate)] / net proceeds of issue

= 6.00(1 Œ 0.25) / 90.00 = 5.0%Ł WACC

is de˚ned by weighting these individual costs by their proportion in the company™s

existing capital structure and summating the products to arrive at their WACC. ˜e simplest

method is to use balance sheet data as follows:

Weighted Average Cost of Capital: Book Value Capital Structure (£ million) Weight Component Cost (%) Weighted Cost (%) Ordinary shares 12 0.50 13.0 6.50 Retained Earnings 4 0.17 12.1 2.06 Preference Shares 2 0.08 15.0 1.20 Debentures 6 0.25 5.0 1.25 Totals 24 1.00 11.01 Download free eBooks at bookboon.com

74 Capital Gearing and the Cost of Capital

However, this approach invites criticism. Although the capital mix will not change,

book

weights have

been applied to component costs when clearly

market values

relating to current additions to the capital

structure are more appropriate. What is required for

new

investment is a weighted average of its

marginal

costs of capital and not

historical

costs.

Weighted Average Cost of Capital: Market Value Capital Structure (£ million) Weight Component Cost (%) Weighted Cost (%) Ordinary Shares 84.0 0.89 13.0 11.57 Retained Earnings 4 0.04 12.1 0.48 Preference Shares 0.8 0.01 15.0 0.15 Debentures 5.4 0.06 5.0 0.30 Totals 94.2 1.00 12.50 ˜e substitution of market values for book values in our WACC calculation raises the company™s discount

rate from 11.01 percent to 12.5 percent.

2. Project Evaluation

Project viability is established by applying the NPV decision rule to the project data, using the12.5 per

cent WACC based on market values as the cut-o˛ rate. ˜e NPV of the £100k investment yielding £11.5k

in perpetuity is given by:

NPV = [(11,500 / 0.125) = £92,000] Œ £100,000 = (£8,000)So, the project

under-recovers

and should be

rejected

. However, it is worth noting that if we had applied

book values to WACC the project would appear acceptable.

NPV = [(11,500 / 0.1101) = £104,450] Œ £100,000 = £4,450Even so, you will be in no doubt as to which decision is correct. If wealth is to be maximised, projects

must always be evaluated in terms of current investment opportunities foregone. Hence, the market

value of capital employed and its corresponding incremental yield are the correct factors to determine

a ˚rm™s WACC as an overall cut-o˛ rate for investment.

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75 Capital Gearing and the Cost of Capital

3. Shareholder Wealth Maximisation

˜e shareholder wealth implications of the correct accept-reject decision using WACC as a discount rate

can be con˚rmed by analysing the investment™s impact on the equity yield. Using market weights from

the previous table, let us ˚rst calculate the proportion of equity applied to the investment:

£100,000 (0.890) = £89,000Next calculate the annual cash return available to the new ordinary shareholders.

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76 Capital Gearing and the Cost of Capital

Finally, let us reformulate this cash return as the investment™s yield on the ordinary share issue. Capital Investment £ Capital Cost % Investor Return £ Annual Cash Inflow 11,500 Retained Earnings £100,000 x 0.04 4,000 12.1 484 Preference Shares £100,000 x 0.01 1,000 15.0 150 Debentures £100,000 x 0.06 6,000 5.0 300 11,000 934 Ordinary Shares 10,566 Finally, let us reformulate this cash return as the investment™s yield on the ordinary share issue.

Project equity yield = £10,566 / £89,000 = 11.87%

Because this is less than the 13 per cent marginal cost of new issues calculated at the outset of our

analysis, we can con˚rm that the investment proposal should be rejected.

You may also care to verify that even if the 12.1 per cent cost of retained earnings were incorporated

into the yield calculation to provide a more comprehensive measure of the equity rate (i.e. dividends

plus retentions) the overall return would only be 11.88 percent.

Since this too, is lower than the 12.1 percent yield on shares currently in issue, the project should still

be ignored.

6.4 Summary and Conclusions

˜e previous Activity serves as a timely reminder that eˆcient ˚nancial management (based

on agency theory) should comprise two distinct but inter-related functions as shareholder wealth

maximisation criteria.

Ł ˜e

investment decision

, which identi˚es and selects opportunities to maximise ENPV.

Ł ˜e

˝nance decision

, which identi˚es potential fund sources required to sustain investment,

evaluates the return expected by each, then selects the optimum mix that minimises their

overall combined cost (WACC).

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77 Capital Gearing and the Cost of Capital

However, as we mentioned in Chapter One, the derivation of an optimal capital structure and minimum

WACC is a controversial subject. So far, we have observed that the issue of lower-cost debt (which might

also incorporate tax relief) rather than equity, should reduce WACC and increase overall corporate value.

But as we shall explain in Chapter Seven, this

may only

be

true up to a point

.What is now termed a ﬁtraditionalﬂ view of ˚nance states that when debt is introduced into a ˚rm™s

capital structure, leverage may initially reduce WACC and increase total corporate value. However, when

shareholders and debt ˚nanciers perceive that gearing levels are excessive, the WACC will increase and

value fall. ˜is

saucer-shaped

WACC plotted against increasing leverage is caused by a combination

of higher returns required for existing equity issues and higher interest rates on new increments of

debt. It compensates both capital providers for the higher

˝nancial risk

associated with their respective

corporate investment.

Because companies have a contractual obligation to pay interest on debt, any variability in earnings

arising from business risk is transferred to the shareholders. As ﬁlenders of last resortﬂ they bear the

inconsistency of investment returns, which may result in a reduction, or even non-payment, of dividends.

˜is ﬁpecking orderﬂ phenomenon is ampli˚ed as the gearing ratio rises.

To compensate for higher levels of ˚nancial risk, rational shareholders require higher yields on their

holdings, thereby producing a lower capitalised value of expected earnings available for distribution

and. a lower share price. At extremely high levels of gearing, the situation may be further aggravated

by debt holders. ˜ey too, may require ever-higher rates of interest as their investment takes on the

characteristics of equity. It no longer represents a

prior

claim, but perhaps the

only

claim, on either the

˚rm™s income or assets.

To summarise this ﬁtraditionalﬂ approach to theories of capital structure, WACC and corporate value:

Beyond some minimum point, incremental borrowing will not reduce WACC. It increases because of the detrimental

e˛ect on existing equity prices, thereby increasing shares yields. In turn, this leads to higher marginal costs of debt and

lower prices for further increments of borrowing, resulting in a dramatic decline in the overall value of the ˜rm.

So far so good, but a contrary view originally published by Modigliani and Miller (MM) in 1958

hypothesised that WACC and total corporate value remain constant, irrespective of the level of

gearing (leverage).

Equity and debt returns (like dividends and retained earnings) are

perfect economic substitutes

. With the introduction

of cheaper debt ˜nance, any rational change in the gearing ratio immediately elicits a compensatory change in the

cost of equity. Speci˜cally, it o˛sets changes in the level of ˜nancial risk, thereby leaving WACC and overall corporate

value the same.

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78 MM and Capital Structure

If you are perplexed, or feel we are moving too fast, don™t worry. From 1958 through to the late 1970s the

traditional academic community and ˚nancial analysts™ perception of leverage dynamics were thrown

into disarray by the application of MM™s ﬁlaw of one priceﬂ, creating a ˙urry of new research.

But as we shall discover in Chapter Seven, their basic hypothesis is ﬁlogical if you think about itﬂ.

Ł In terms of the investment decision, WACC still occupies a pivotal position as an opportunity

cost (return) criterion which justi˚es the ˚nance decision.

Ł A company wishing to maximise shareholders™ wealth would still only deploy funds if their

marginal yield at least satis˚es the rates of return that its investors can earn elsewhere at

commensurate risk.

˜erefore, the two ˚nal questions we must answer in Part ˜ree are ﬁwhyﬂ and ﬁhowﬂ MM could

theoretically discredit traditional approaches to capital structure, WACC and corporate value?

6.5 Selected Reference

Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 3, June 1958.

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79 MM and Capital Structure

7 MM and Capital Structure

Introduction

For the purpose of exposition, the derivation of a company™s weighted average cost of capital (WACC)

in Chapter Six was kept simple. Given ˚nancial management™s strategic objective is to maximise the

market value of ordinary shares (common stock) our analysis assumed that:

-˜e value attributed by the market to any class of ˚nancial security (debt or equity) is the

expected PV of its cash returns, discounted at an opportunity rate that re˙ects the ˚nancial

risk associated with those returns.

-˜e ENPV of a project, discounted at a company™s WACC (based on debt plus equity) is the

amount by which the market value of the company will increase if the project is accepted;

subject to the constraint that acceptance does not change WACC.

We speci˚ed

three

necessary conditions that underpin this constraint and justify the use of WACC as a

cut-o˛ rate for investment.

-˜e new project has the same business risk as the company™s existing investment portfolio.

-˜e project is small, relative to the scale of its existing operations.

-˜e company intends to retain its existing capital structure (i.e. ˚nancial risk is constant).

Yet, we know that even if business risk is

homogenous

and projects are

marginal

, capital structures and

the ˚nancial risk of future investments are rarely

stable

. ˜e component costs of corporate ˚nance

(and hence WACC) are susceptible to periodic change as

external

forces unfold. ˜e availability of new

external funding may also be a

limiting factor

, as evidenced by the market™s aversion to provide debt or

equity a˝er the 2007 global meltdown.

So, let us develop a

dynamic

critique of capital structure and the overall cost of capital (WACC) and ask

ourselves whether management can increase the value of the ˚rm, not simply by selecting an optimal

investment

, but also by manipulating its

˝nance

. If so, there may be an optimal capital structure arising

from a debt-equity

trade-o˜

, which elicits a least-cost combination of ˚nancial resources that minimises

the ˚rm™s WACC and maximises its total value.

In the summary to Chapter Six, we touched on the case for and against an optimal capital structure and

WACC based on ﬁtraditionalﬂ theory and the MM economic ﬁlaw of one priceﬂ respectively. Later in this

Chapter we will pick up on these con˙icting analyses in detail. Speci˚cally, we shall examine the MM

arbitrage

proof (1958) whereby investors can pro˚tably trade securities with di˛erent prices between

companies with di˛erent leverage until their WACC and overall value are in

equilibrium

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80 MM and Capital Structure

Unlike the traditionalists, MM maintain that the equilibrium value of any company is

independent

of its

capital structure and derived by capitalising expected project returns at a

constant

WACC appropriate

to their class of business risk. Yet both theories begin with a common assumption. Because of higher

˚nancial risk, the cost of equity is higher than the cost of debt and rises with increased leverage (gearing).

So, before we analyse why the two theories part company, let us initially explain how increased gearing

a˛ects shareholder returns by graphing the relationship between earnings yields and net operating

income (NOI)

i.e. earnings before interest and tax (EBIT) when ˚rms incorporate cheaper debt into

their capital structure.

Like our approach to the MM dividend debate in Chapter Four, we shall underpin their theoretical

exposition with appropriate Activities (and outline solutions). And because of the subject™s complexity,

we shall (again) develop a data set using a Review Activity to summarise and critique our analysis as a

guide to further study.

7.1

Capital Structure, Equity Return and Leverage

To assess the impact of a changing capital structure on capital costs and corporate values, let us begin

with a fundamental assumption of capital market theory, which you ˚rst encountered in Chapter One,

namely that investors are

rational

and

risk averse

. Companies must o˛er them a return, which is inversely

related to the probability of its occurrence. ˜us, the crucial question for ˚nancial management is whether

a combination of stakeholder funds, related to the earnings capability of the ˚rm, can minimise the

risk that confronts each class of investor. If so, a ˚rm should be able to minimise its own discount rate

(WACC) and hence, maximise total corporate value for the mutual bene˚t of all.

We know from previous Chapters that

total

risk

comprises two inter-related components with which you

are familiar,

business risk

and

˝nancial risk

. So, even when a ˚rm is ˚nanced by equity alone, the pattern

of shareholder returns not only depends upon periodic post-tax pro˚ts (business risk). It also arises

from managerial decisions to withhold dividends and retain earnings for reinvestment (˚nancial risk).

As we explained in Chapter ˜ree, using the Gordon growth model (1962) if rational, risk averse investors

prefer dividends now, rather than later, the question arises as to whether their equity capitalisation rate is

a positive function of a ˚rm™s retention ratio. In otherwords, despite the prospect of a capital gain, does

a ﬁbird in the handﬂ philosophy require a premium for the ˚nancial risk associated with any diminution

in the dividend stream? If so, for a given

investment

policy, corporate

˝nancial

policy must also a˛ect

the overall discount rate that management applies to NPV project analyses and therefore the market

value of ordinary shares.

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81 MM and Capital Structure

When a ˚rm issues debt, we can apply the same logic to arrive at similar conclusions. Financial policies

matter

because the degree of leverage arising from the debt-equity ratio (like the dividend payout ratio)

determines the level of ˚nancial risk that confronts the investor.

˜e ˜eoretical Background

Initially, when a ˚rm borrows, shareholder wealth (dividends, plus capital gains) can be increased if

the e˛ective cost of debt is lower than the original earnings yield. In eˆcient capital markets such an

assumption is not unrealistic:

Ł Debt holders receive a guaranteed return and in the unlikely event of liquidation are usually

given security in the form of a prior charge over the assets.

Ł From an entity viewpoint, debt interest quali˚es for tax relief.

You should note that the productivity of the ˚rm™s resources is unchanged. Irrespective of the ˚nancing

source, the same overall income is characterised by the same degree of business risk. What has changed

is the mode of ˚nancing, which increases the investors™ return in the form of earning per share (EPS) at

minimum ˚nancial risk. So, if this creates demand for equity and its market price rises proportionately,

the equity capitalisation rate should remain

constant

. For the company, the bene˚cial e˛ects of cheaper

˚nancing therefore outweigh the costs and as a consequence, its overall cost of capital (WACC) falls

and total market value rises.

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82 MM and Capital Structure

Of course, the net bene˚ts of gearing cannot be maintained inde˚nitely. As a ˚rm introduces more

debt into its capital structure, shareholders soon become exposed to greater ˚nancial risk (irrespective

of dividend policy and EPS), even if there is no likelihood of liquidation. So much so, that the demand

for equity

tails o˛ and its price begins to fall, taking total corporate value with it. At this point, WACC

begins to rise.

˜e increased ˚nancial risk of higher gearing arises because the returns to debt and equity holders

are

interdependent

stemming from the same investment. Because of the contractual obligation to pay

interest, any variability in operating income (EBIT) caused by business risk is therefore transferred to the

shareholders who must bear the inconsistency of returns. ˜is is ampli˚ed as the gearing ratio rises. To

compensate for a higher level of ˚nancial risk, shareholders require a higher yield on their investment,

thereby producing a lower capitalised value of earnings available for distribution (

i.e. lower share price).

At extremely high levels of gearing, the situation may be further aggravated by debt holders. ˜ey too,

may require ever-higher rates of interest as their investment takes on the characteristics of equity and

no longer represents a prior claim on either the ˚rm™s income or assets.

Even without increasing the interest rate on debt, the impact of leverage on shareholder yields can be

illustrated quite simply. Consider the following data (£ million):

Company Ulrich plc Hammett plc MARKET VALUES Equity 100 60 Debt _ 40 Total 100 100 NET OPERATING INCOME Norm Deviations Norm Deviations EBIT 8.0 10.0 12.0 8.0 10.0 12.0 Interest (10%) 4.0 4.0 4.0 EBT 8.0 10.0 12.0 4.0 6.0 8.0 Corporation Tax (25%) 2.0 2.5 3.0 1.0 1.5 2.0 EAT 6.0 7 .5 9.0 3.0 4.5 6.0 Earnings Yield (%) 6% 7.5% 9% 5% 7.5% 10% Download free eBooks at bookboon.com

83 MM and Capital Structure

˜e two companies (Ulrich and Hammett) are identical in every respect except for their methods of

˚nancing. Ulrich is an all-equity ˚rm. Hammett has £40 million of 10 percent debt in its capital structure.

A comparison of net operating income (EBIT) and shareholder return (earnings yield) is also shown, if

business conditions deviate 20 percent either side of the norm.

What the table reveals is that the returns to ordinary shareholders in the all-equity company only ˙uctuate

between 6 percent and 9 percent as EBIT (business risk) ˙uctuates between £8 million and £12 million.

However, the existence of a ˚xed interest component for the geared company ampli˚es business risk in

terms of the total risk borne by the ordinary shareholder. Despite the bene˚ts conferred on Hammett

and its shareholders by the tax deductibility of debt, the greater range of equity returns (5Œ10 per cent)

implies greater ˚nancial risk.

˜us, if shareholders act rationally and business prospects are poor, they may well sell their holdings

in the geared company, thereby depressing its share price and buy into the all-equity ˚rm causing its

price to rise.

Our preceding discussion suggests that for a given level of earnings, a company might be able to trade

the costs and bene˚ts of debt by a combination of fund sources, which achieves a lower WACC and

hence a higher value for equity. To implement this strategy, however, management obviously need to be

aware of shareholder attitudes to its existing ˚nancial policy and

those of competitors under prevailing

economic conditions. Even ﬁblue chip ﬁcompanies with little chance of liquidation are not immune to

˚nancial risk.

Activity 1

Use the previous data for Ulrich and Hammett to:

1. Graph the relationship between their respective earnings yield (vertical axis) and EBIT (horizontal axis) and

establish the indi˛erence point between their shareholder clienteles.

2. Summarise your graph™s illustrations concerning shareholder preferences.

From the raw data you should have observed that if shareholders require a 7.5 per cent return and

the EBIT (NOI) of both companies equals £10 million, they would be indi˛erent to investing in either,

irrespective of current ˚nancial policies. By plotting a graph, however, you can also see that the relationship

between earnings yield and EBIT is

positive

and

linear

for both companies but

di˜erent

. For the all-

equity ˚rm it is less severe, with a shareholder™s return of zero corresponding to an EBIT ˚gure of zero

that passes through the origin in Figure 7.1. For the geared company, the EBIT ˚gure equivalent to a

zero earnings yield intersects the horizontal axis at the value of 10 percent debenture interest payable

(£4 million) and rises more steeply.

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84 MM and Capital Structure

EBIT (£ million) 8 12 10 4 0 9.0 6.0 7.5 5.0 10.0 Earnings Yield % Ulrich (Ungeared) Hammett (Geared) Figure 7.1:

Capital Gearing and the Relationship between EBIT and Earnings Yield

˜e intersection of the two straight lines represents the point of

indi˜erence

between the two companies.

To the le˝ of this point, shareholders would prefer to invest in Ulrich (ungeared) since they receive a

better return for a lower level of EBIT. To the right, they would prefer Hammett (geared) for the same

reasons. What we are observing is that leverage, which here means the incorporation of 10 percent loan

stock into a ˚rm™s capital structure, increases shareholders™

sensitivity

to changes in EBIT (business risk)

and therefore the ˚nancial risk associated with equity; hence the steepness of the line.

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85 MM and Capital Structure

7.2 Capital Structure and the Law of One Price

˜e previous Activity illustrates why rational, risk adverse investors prefer the ordinary shares of higher

geared companies when economic conditions are good or improving but switch to lower geared ˚rms

when recession looms. Both strategies represent an

optimum

risk-return trade o˛ because:

Ł Ordinary shares represent a more speculative investment when there is an increasing contractual

obligation on the part of the company to pay periodic interest on debt.

Ł ˜e higher the gearing and more uncertain a ˚rm™s overall pro˚tability (EBIT) the greater the

˙uctuation in dividends plus reserves.

As mentioned earlier, the returns to debt and equity holders are

interdependent

. ˜ey stem from the

same resources. So, what we have observed is the transfer of business risk to shareholders who must

bear the inconsistency of returns as the ˚rm gears up. ˜us, it would seem that management should

˚nance corporate investments so that their shareholders (to whom they are ultimately responsible)

receive the highest return for a given level of earnings and risk. And this is where MM disagree with

traditional theorists.

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86 MM and Capital Structure

˜e Traditional ˜eory of Capital Gearing and WACC

Traditionalists believe that if a ˚rm substitute™s lower-cost debt for equity into its capital structure, WACC

will fall and value rise to a point of indebtedness where both classes of investor will require higher returns

to compensate for increasing ˚nancial risk. ˜erea˝er, WACC rises and value falls, suggesting an optimum

level of gearing that minimises WACC and maximises value. Figure 7.2 sketches these phenomena.

˜e debt-equity ratio (V

D/VE) is plotted along the horizontal axes of both diagrams. ˜e costs of both

types of capital (setting K

d < Ke) are given on the vertical axis of the upper graph. ˜e vertical axis on

the lower graph plots total market value (V=V

E +VD). To keep the analysis simple K

d is held constant

and its tax deductibility is ignored. Our aim is not to develop a real world model (more of which later)

but to illustrate the basic relationships between capital costs, corporate value and leverage.

Maximum 0 VD/VV 0 Minimum VD/VE Kd K Ke % K i £ V Figure 7.2:

Traditional Theory with a Constant Cost of Debt in a Taxless World

Figure 7.2 con˚rms the traditional view that WACC is characterised by a U-shaped average cost curve

K (familiar to economists).˜is is because the bene˚ts of cheaper debt ˚nance (K

d<Ke) are eventually

o˛set by the cost of equity, which increases

exponentially

as the ˚rm gears up.

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87 MM and Capital Structure

Turning to total market value V, (equity plus debt) if we de˚ne the relationship:

(19) V = I / KWhere:

I = NOI = net operating income (earnings before interest)

: V = VE + VD = total market value

VE = market value of equity

VD = market value of debt

K = WACC,

= Ke (VE / VE + VD) + Kd (VD / VE + VD)

= Ke (VE / V) + K

d (VD / V) Ke = cost of equity

Kd = cost of debt

We now observe that an

inverse

relationship exists between V and K, given I (NOI). As one rises, the

other falls and

vice versa.

˜us, the lower graph of Figure 7.2 illustrates that the degree of leverage,

relative to the total market value of the ˚rm, has an

inverted

U-shaped function. As K (WACC) responds

to changes in the gearing ratio and the rising cost of equity, V presents us with a mirror image. So,

according to traditional theory, if companies borrow at an interest rate lower than their returns to equity,

the implications for ˚nancial management are clear.

For a given investment policy, there exists an optimal ˜nancial policy (debt-equity ratio) which de˜nes a least-cost

combination of ˜nancial resources.

At the point where overall cost of capital (WACC) is minimised, total corporate value is maximised and so too, is the

market value of ordinary shares.

˜e MM Cost of Capital Hypothesis

Like much else in ˚nance, the traditional case for an optimal capital structure did not arise from

empirical evidence: merely a plausible assumption concerning the cost of equity at di˛erent levels of

gearing (initially constant, then rising with greater rapidity). But what if the relationship between the

two is mistaken? Would an optimal WACC and corporate value still emerge?

To answer both these questions, MM (

op.cit

.) developed an alternative hypothesis, which produced two

conclusions that confounded both traditional theorists and ˚nancial analysts.

Ł ˜e total value of a ˚rm represented by the NPV of an income stream discounted at a rate

appropriate to its business risk, should be una˛ected by shi˝s in ˚nancial structure.

Ł Any rational debt-equity ratio should produce the same cut-o˛ rate for investment (WACC).

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88 MM and Capital Structure

Unlike many of their contemporaries, MM based their conclusions on

partial equilibrium

analysis

(not anecdotal evidence) prefaced by a number of rigorous assumptions, which they later relaxed to

incorporate subsequent empirical research. ˜ese should be familiar, since they are based on

perfect

markets

and Fisher™s Separation ˜eorem (1930) outlined in Chapter One

-Investors are rational.

-Information is freely available.

-Transaction costs are zero.

-Debt is risk-free (the return is constant)

-Investors are indi˛erent between corporate and personal borrowing.

-˜e tax system is neutral.

MM also based their analysis on the traditional equation for total market value:

(19) V = I / KHowever, where they disagree with traditional theory relates to their de˚nition of WACC, which hinges

on the behaviour of the equity capitalisation rate as a ˚rm gears up.

MM reason that WACC re˙ects the business risk associated with the variability of total earnings, rather

than their ˚nancial risk,

i.e. how they are packaged for distribution in the form of interest and dividends.

˜ey maintain that irrespective of the debt-equity ratio (V

D/VE) if expected earnings (I) remain the same,

then WACC (K) and hence total value (V) must also be constant.

Based on their ﬁeconomic law of one priceﬂ MM further reasoned that irrespective of leverage, close

˝nancial substitutes

, such as similar companies cannot sell at di˛erent prices. Two companies with the

same business risk and identical total income will have the same total market value and WACC, even

if their gearing ratios di˛er.

Expressed algebraically, if:

V1 = V2 = the value for two companies.

I1 = I2 = average NOI represented by the expected value of its probability distribution

˜en the WACC for any company in the same risk class:

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89 MM and Capital Structure

And because K =

Ke in the unlevered ˚rm, the WACC for the geared company must also equal the cost

of equity capital K

e of the all equity ˚rm.

˜us, if the cost of debt K

d is constant (an assumption that MM later relax) all that needs to be resolved

is the precise relationship between the rising cost of equity K

e and the debt-equity ratio V

D/VE when a

˚rm gears up. Is it

exponential

, as the traditionalists suggest (Figure 7.2) or not.

VD/VKd K Ke % K i 0 V VD/V0 £ V Figure 7.3:

The MM Theory with a Constant Cost of Debt in a Taxless World

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90 MM and Capital Structure

According to MM, if we ignore corporate taxation and tax relief on interest, the equity capitalisation

rate K

e will still increase but not exponentially as the traditionalists believe. ˜e rise

exactly o˜sets

the

bene˚ts of increasing the proportion of cheaper loan stock in a ˚rm™s capital structure leaving WACC

unchanged. ˜is

linear

relationship is sketched in the upper graph of Figure 7.3, which translates into

the following equation.

(21) Keg = Keu + [(VD / VE) ( Keu Œ Kd)]Where:

Keg = the cost of equity in a geared company

Keu = the cost of equity in an ungeared company

Kd = the cost of debt capital

VD = the market value of debt in the geared company

VE = the market value of equity in the geared company

Ł Keg (leveraged) is equivalent to K

eu, the capitalisation rate for an all-equity stream of the same

class of business risk, plus a premium related to ˚nancial risk. ˜is is measured by the debt-

equity ratio (V

D/VE) multiplied by the spread between K

eu and K

d.Ł ˜e ˚nancial risk premium (the second term on the right of the equation) causes equity yields

to rise at a

constant

rate as compensation for ˚nancial risk when the ˚rm gears up.

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91 MM and Capital Structure

Since the WACC in companies of equivalent business risk is the same, irrespective of leverage, their

total market value (V) will also be the same if the companies are identical in every respect except their

gearing ratio. ˜us:

(22) VU = VG = VE +VDWhere:

VU = the market value of an ungeared all equity company

VG = market value of an identical geared company (equity plus debt)

˜e lower graph of Figure 7.3 plots constant value (V) against an increasing debt-equity ratio (V

D/VE).If WACC and overall corporate value are una˛ected by leverage as MM hypothesise, the implication for

strategic ˚nancial management are profound. As we ˚rst mentioned in Chapter One, ˚nancial decisions

(which include the dividend policy, as well as gearing) are irrelevant to investment decisions (the valuation

of capital projects and their selection).

Activity 2

In a subject still dominated by the work of Modigliani and Miller it is important that you refer to their original articles,

if only to con˜rm what you read elsewhere.

MM™s 1958 paper ﬁThe Cost of Capital, Corporation Finance and the Theory of Investmentﬂ (referenced at the end of

this Chapter) sets out their original case for the irrelevance of ˜nancial structure to corporate valuation and capital cost

(WACC) in a perfect capital market. Search for it and skim through, to get the broad thrust of their arguments (even if

you ˜nd the mathematics complex). Then produce brief answers to the following questions.

a. There are three

propositions

advanced by MM. What are they and how are they proved?

b. How do MM™s conclusions di˛er from a traditional view of capital structure in a taxless world where the cost

of debt is constant?

c. Within the context of investment appraisal, what are the implications of MM™s hypothesis for ˜nancial

management?

a) ˜e Propositions

Using our own notation, the three propositions advanced by MM are:

Proposition I:

Overall market value (V) is

independent

of the debt-equity ratio (V

D/VE).Proposition II:

To o˛set ˚nancial risk, the equity capitalisation rate (K

eg) increases at a

constant

rate as

VD/VE rises, with two corollaries:

-K is una˛ected by V

D/VE , -K = Keu for an unlevered ˚rm.

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92 MM and Capital Structure

Proposition III:

Shareholder wealth is maximised by

substituting

an equity capitalisation rate of an

unlevered ˚rm (K

e u), for the cut-o˛ rate (K) of a levered ˚rm.

MM then explain how:

i. Proposition I can be proved by

arbitrage

(more of which later).

ii.

Proposition I can be used to prove Proposition II, which states that K is una˛ected by

VD/VE. iii.

Proposition III follows logically from Propositions II and III, since market value equals equity

value (V = V

E) and therefore K = K

e in an unlevered ˚rm.

b) MM™s Conclusions

Even in a world of zero taxation with a constant cost of debt, a comparison of Figure 7.2 with Figure 7.3

reveals that MM™s conclusions contrast sharply to a traditional view.

WACC does not vary with gearing. ˜ere is no optimal debt-equity ratio and the market value of the

˚rm remains constant. According to MM, the cost of equity capital is no longer an

exponential

function

of increasing leverage. Given MM™s contention that K is constant, K

e rises

linearly

as V

D/VE increases.

c) ˜e Investment Implications

If MM™s hypothesis is correct, the ﬁtraditionalﬂ ˚nancial decisions that confront management when

investment decisions include debt are eliminated. ˜e net result is that WACC (the cut-o˛ rate for

investment) and total corporate value remain the same. Gearing is therefore irrelevant to project

evaluation and shareholder wealth maximisation.

7.3 MM and Proposition I (the Arbitrage Process)

Your reading for Activity 2 should con˚rm how the logic of MM™s cost of capital hypothesis stems from

their ˚rst proposition that corporate value is independent of capital structure based on

arbitrage

and

partial equilibrium

analysis.

Arbitrage occurs when investors sell ˜nancial securities to buy cheaper perfect substitutes, thereby depressing the price

of the former and increasing the price of the latter, until their market prices are in equilibrium.

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93 MM and Capital Structure

In perfect markets, MM maintain that if a traditional view of capital structure were to exist, it should only

be a short-run

dis-equilibrium

phenomenon. Rational (risk-averse)

arbitrageurs

will respond quickly to

prevent the existence of two ˚rms with identical business risk and the same expected NOI from selling

at di˛erent prices.

Ł Shareholders in an over-valued company (what traditionalists de˚ne as highly geared) will

change its total value by selling shares in that company and buying shares in an under-valued

(i.e.ungeared) company. To implement these transactions, shareholders will even undertake

personal borrowing to maximise their stake in the ungeared company, up to a point where

their personal investment portfolios have the same degree of leverage as the overvalued ˚rm.

Ł As a result of what MM term

home-made

leverage

(personal borrowing), investor income is

increased at no greater ˚nancial risk. Eventually, through supply and demand, the price of

shares in the overvalued company will fall, while that of the undervalued company will rise

until no further ˚nancial advantage is gained. At this point of

equilibrium

, overall market value

and the overall cost of capital (WACC) for the two companies will also be the same.

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94 MM and Capital Structure

˜e Mathematics of Arbitrage

Before illustrating the arbitrage process with a numerical Activity, let us express it algebraically.

Given what MM term

disequilibrium

: a temporary phenomenon where the total market value of an

ungeared company (V

1) is lower than that of a geared company (V

2), we can de˚ne:

(23) V1 = VE1 < V2 = VE2 + VD2Now assume a shareholder holds a proportion A(V

E2) of the total equity in the geared company. ˜e

investment will yield:

(24) Y2 = A(I Œ Kd . VD2)Where:

Y2 = the income available to the shareholder

I = I1 = I2 = NOI

Kd = the interest on corporate debt.

˜e investor now sells his shares in Company 2 for A(V

E2) and also borrows an amount equal to A(V

D2) at the same rate of interest K

d to invest A(V

E2 + VD2) in the ungeared Company 1.

He is therefore substituting

personal

leverage for

corporate

leverage by borrowing the same proportion

(AV

D2 / AVE2) as that of the geared company™s debtŒequity ratio (V

D2 / VE2).˜is new holding in Company 1 will yield Y

1 (net of interest on personal borrowing):

(25) Y1 = {A ( VE2 + VD2 ) I } - Kd .VD2 V1 = A (V2 .I - Kd . VD2 ) V1 And further simplifying:

(26) Y1 = A (V2 . Y2) / V1 ˜us, we observe that if V

1 < V2, as the traditionalists advocate, then Y

1 > Y2 and shareholders™ income

can be increased by arbitrage.

However, as

MM suggest, switching from the geared (overvalued) company to the all-share ˚rm will

eventually depress the equity value of the former, while raising the price of the latter, until they are

in equilibrium.

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95 MM and Capital Structure

At this point, (where V

1 = V2 and Y

1 = Y2) the arbitrage process elicits no further gains and shareholders

will be indi˛erent to levels of gearing. Moreover, because we have assumed that NOI (I) is identical for

both companies, it also follows that their WACC (K) must be identical.

To summarise MM™s basic theoretical proposition.

Given: I = I

1 = I2Ł when V

1 < V2 : Y1 > Y2 and K

1 > K2Ł when V

1 = V2 : Y1 = Y2 and K

1 = K2˜us, they conclude that in equilibrium under certain conditions, changes in capital structure can have

no e˛ect on overall shareholder income, corporate value, or cost of capital.

7.4 MM and Real World Considerations

We could now move on from Proposition I to prove Propositions II and III and then extend MM™s analysis

into a real world of di˛erential corporate and personal taxation, where the cost of debt is tax deductible

and no longer constant. However, we shall save all this for the Activities in our companion Exercise text.

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96 MM and Capital Structure

For the moment, we simply need to appreciate that the MM arbitrage proof confounded the traditional

academic and investment community, who argued that their assumption of perfect markets, particularly

a neutral tax system without tax relief on debt interest invalidated their conclusions. However, MM

(1963) were the ˚rst to concede that an allowance for tax relief will reduce the cost of loan stock, lower

WACC and increase total value as a ˚rm gears up. But this is a result of ˚scal policy and not business or

˚nancial policy. ˜e whole point of their hypothesis was to provide a

benchmark

to assess the impact of

introducing more realistic assumptions as a basis for more complex analyses (which we shall evaluate

in our Exercise text) such as:

-Do personal as well as corporate ˚scal policies a˛ect capital structure?

-Are corporate borrowing and investment rates equal?

-How do investor returns (debt and equity) behave with extreme leverage?

-Are management better informed than stock market participants?

-Do managerial objectives con˙ict with those of investors?

-And if so, do management prefer di˛erent sources of ˚nance?

Review Activity

Elbow (ungeared) Dimebag (geared) Distributable Earnings (No Tax) NOI (I) 100 = 100 Debt Interest (Kd = 5%) - 10 Shareholder Income 100 > 90 Market Values Equity (VE) 1000 > 900 Debt (VD) - 200 Total Value (V) 1000 < 1,100 Capital Costs Equity Yield (Ke) 10% = 10% Cost of Debt (Kd) - 5% WACC (K = I / V) 10% > 9.09% ˜e previous table presents a series of

traditional

˚nancial relationships between two ˚rms (Elbow and

Dimebag) that are identical in every respect, except for their capital structure (• 000).

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97 MM and Capital Structure

Required:

1. Use the data set to illustrate the bene˚ts of arbitrage for an investor who currently owns 10

per cent of Dimebag™s shares.

2. Summarise the e˛ects of arbitrage as more investors enter the process.

An Indicative Outline Solution

From the data you should have observed what MM term

disequilibrium

. ˜e total market value and

WACC of two equivalent companies di˛er. So, arbitrage is a pro˚table strategy for all investors in the

geared ˚rm.

1. ˜e Arbitrage Process

Now let us consider a series of arbitrage transactions for a single investor who holds 10 percent of

the equity in Dimebag (the higher valued geared ˚rm) whose annual income is therefore •9,000

(•90,000 × 0.10).1. She sells her total shareholding for •90,000 (10 percent of •900,000) which reduces the ˚nancial

risk of investing in the geared company to zero.

2. She now buys shares in Elbow (the ungeared, all-equity ˚rm) but how much should she spend?

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98 MM and Capital Structure

3. In order to compare like with like, it is important to hold the investor™s exposure to ˚nancial

risk at the same level as her original investment in Dimebag. With a •90,000 equity stake in

that company, management presumably used this as collateral to borrow •20,000 of corporate

debt on her behalf (i.e.10 per cent of •200,000). So, in a perfect capital market where private

investors can borrow on the same terms as the company, she can substitute

homemade

leverage

for

corporate

leverage to ˚nance her new investment in the all-equity ˚rm.

4. She borrows •20,000 at 5 percent per annum, an amount equal to 10 percent of the ˚rm™s debt.

5. As a result, the investor now has a total of •110,000 (•90,000 cash, plus •20,000 of personal

borrowing) with which to purchase the ungeared shares in Elbow.

6. Because Elbow™s yield is 10 percent, the investor will receive an annual return of •11,000

(•110,000 × 0.10). However, she must pay annual interest on her personal loan (•20,000 × 0.05 =

•1000).˜erefore, her annual net income will be •10,000 (•11,000 Œ •1000).

So, to conclude: is our investor better o˛?

We can measure her change in income as follows:

˜e Arbitrage Process

•

Shareholder income in Elbow (ungeared)

11,000Shareholder income in Dimebag (geared)

9,000Change in income

2,000Interest on borrowing (5%)

1,000Net Gain from

Arbitrage

1,000˜us, shareholder income has increased with no change in ˚nancial risk. ˜e reason the investor has

bene˚ted is because the leveraged shares of Dimebag are overvalued relative to those of Elbow. If proof

be needed, you should be able to con˚rm that the equity capitalisation rates for both ˚rms originally

equalled 10 per cent, despite di˛erences in their total shareholder income.

2. Summary

As more investors enter the arbitrage process (trading shares to pro˚t from disequilibrium) the equity

value of geared ˚rms will fall, whilst those of their ungeared counterparts will rise. To similar but

opposite e˛ect, their equity capitalisation rates will rise and fall respectively, until their overall cost of

capital (WACC) is equal. ˜us, MM™s message to ﬁtraditionalistsﬂ is clear.

In

equilibrium, shareholders will be

indi˚erent

to the degree of leverage and the arbitrage process becomes

irrelevant

to

management™s strategic evaluation of project investment and its wealth maximisation implications.

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99 MM and Capital Structure

7.5 Summary and Conclusions

We have considered whether companies can implement

optimum

˚nancial policies concerning their

capital structure, which

minimise

the weighted average cost of capital (WACC) and

maximise

total

corporate value.

Given your knowledge of capital market eˆciency (Part One), equity valuation (Part Two) and the cost

of debt (Part ˜ree: Chapters™ Five and Six) we have also focussed upon the controversial question as to

whether optimal ˚nancial decisions contribute to optimum investment decisions.

˜e traditional perception is that by trading lower-cost debt for equity, WACC will fall and value rise

to a point of indebtedness where both classes of investor will require higher returns to compensate for

increasing ˚nancial risk. ˜erea˝er, WACC rises and value falls, suggesting an optimum capital structure.

In 1958, Modigliani and Miller (MM) theoretically discredited this view, given the assumptions of perfect

markets with no barriers to trade and tax neutrality, proving that WACC and total value are

independent

of ˚nancial policy. Based on the economic

law of one price

, they used

arbitrage

to demonstrate that close

˝nancial substitutes

, such as two ˚rms in the same class of business risk with identical net operating

income, cannot sell at di˛erent prices; thereby negating ˚nancial risk.

Since MM published their original hypothesis in 1958, the capital structure debate has ebbed and ˙owed

with a surprising lack of consensus among academics, researchers and practitioners. To complicate

matters, subsequent empirical evidence has inevitably focussed on modest (rational) debt equity ratios,

which are the norm, rather than occasional, extreme (irrational) leverage that creates ˚nancial distress

and bankruptcy, such as the 2007 global meltdown.

To learn the lessons of the recent past, perhaps the academic debate must take a new turn. Real world

investors (including corporate management) could also evaluate their past mistakes by reviewing MM™s

basic propositions under extreme conditions. ˜ey provide a sturdy framework for analysis. Moreover,

their cost of capital hypothesis is entirely consistent with their 1961 dividend irrelevancy hypothesis

covered in Chapter Four, for which there is considerable empirical support, (as we shall also discover

in our Exercise companion).

So, for future reference, here is a graph and summary of the basic arbitrage process.

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100 MM and Capital Structure

Minimum Cost Arbitrage Arbitrage Ke (MM) Ki% Ke (MM) Ke (Traditional) K (MM) K (Traditional) Kd VD / VE Kd K Ke (Traditional) Optimum leverage All-Equity 0 Figure 7.4:

A Traditional View of Optimal Capital Costs and MM Equilibrium Proposition I: The Arbitrage E˛ects

According to MM, a traditional view of capital structure (characterised by perfect markets where

companies in the same class of business risk have a di˛erent WACC and overall value) can only be a

temporary phenomenon.

Arbitrageurs

will begin trading and force the two variables into equilibrium.

Figure 7.4 illustrates the e˛ect. Diagrammatically, arbitrage causes both the traditional

U shaped

K curve

(WACC) and the

exponential

Ke curve (cost of equity) to straighten out into

linear

functions.

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101 MM and Capital Structure

To summarise MM™s basic theoretical position:

Ł Corporate value should depend on the

agency principle

(Chapter One) de˚ned by investor-

managerial

agreement on the average level of future earnings and their

variability

(business

risk), rather than the

proportion

distributed (˚nancial risk).

Ł Dividend and retention decisions should be

irrelevant

to the market

price of a share

(Chapter Four).

Ł As a determinant of WACC and total corporate value, the division of returns between debt

and equity should also be

perfect substitutes

.7.6 Selected References

1. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 3, June, 1958.

2. Modigliani, F. and Miller, M.H., ﬁCorporate Income Taxes and the Cost of Capital: A Correctionﬂ,

American Economic Review,

Vol. LIII, No. 3, June, 1963.

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102 Part Four:

The Portfolio Decision

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103 Portfolio Selection and Risk

8 Portfolio Selection and Risk

Introduction

Optimum investment, dividend and ˚nancing decisions implemented by corporate management on

behalf of a multiplicity of shareholders, to whom it is ultimately responsible, ideally require knowledge

of their disparate attitudes toward project risk-return and consumption-wealth preferences. But when

ownership is divorced from control, direct communication between a company and its owners, as well

as other stakeholders and prospective investors, concerning these motivational factors is impractical.

For real world companies with a stock exchange listing in markets characterised by uncertainty, what

management does and what providers of capital desire are ultimately determined by the law of supply and

demand for equity and other ˚nancial securities, measured by movements in their respective prices and

returns. Unfortunately, there is still no uniformity of opinion as to what drives the market. For example,

is it dividends; is it earnings, perhaps pure speculation, or some ˙uctuating combination? Whatever,

management must still allocate the ˚rm™s resources eˆciently between pro˚table investments for the

mutual bene˚t of all stakeholders, which ultimately maximises shareholder wealth, using the market

price of equity as a convenient proxy.

As a benchmark for analyses, this text (like all others in my

bookboon

series) therefore began with an

idealised

picture of market behaviour.

˜e

majority

of investors are rational and risk-averse, motivated by ˚nancial

self-interest

, operating in

reasonably

eˆcient capital markets characterised by a

relatively

free ˙ow of information and

surmountable

barriers to trade.

In Part One

we observed that in a world of certainty, where future events can be speci˚ed in advance,

such investors can con˚dently analyse one course of action relative to another for the purpose of wealth

maximisation.

For an

all-equity

˚rm ˚nanced by ordinary shares (common stock) where the ownership of corporate

assets is divorced from control (the

agency

principle), we de˚ned the

normative

objective of strategic

˚nancial management under conditions of certainty as:

Ł ˜e implementation of optimum investment and ˚nancing decisions using net present value

(NPV) maximisation techniques to generate the highest post-tax money pro˚ts from all a

˚rm™s projects in the form of retentions and distributions. ˜ese should satisfy

existing

owners

(a multiplicity of shareholders) and attract

prospective

equity investors who de˚ne the ˚rm™s

clientele, thereby maximising share price.

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104 Portfolio Selection and Risk

Over their life, individual projects should eventually generate net cash ˙ows that

exceed

their overall

cost of funds to create wealth. ˜is future

positive

net terminal value (NTV) is equivalent to a

positive

NPV, expressed in today™s terms, de˚ned by the project discount rate using the time value of money.

Even when ˚nancial theory moves from a risk-free world to one of uncertainty, where

more than one

future outcome

is possible

, present value (PV) analysis remains the bedrock of rational investment

behaviour. Providing markets are reasonably eˆcient, where all news (good or bad) is soon absorbed

by its participants, investors expect to receive returns discounted at a rate commensurate with their risk.

Taking this

linear

view of society, where ﬁmarkets have no memory and its participants lack perfect

foresightﬂ, we observed that it is possible to de˚ne

expected

investor returns for a given level of risk,

using the techniques of ﬁclassicalﬂ statistical analysis (

Quants

). Assuming a ˚rm™s project cash˙ows (or its stock market returns) are linear, they are

random

variables

that conform to a ﬁnormalﬂ distribution. For every level of risk, there is an investment outcome with

the highest expected return. For every expected return there is an investment outcome with the lowest

expected risk. Using mean-variance analysis, the standard deviation calibrates these risk-return pro˚les

and the likelihood of them occurring, based on probability analysis and con˚dence limits. Wealth

maximisation equals the maximisation of investor

utility

using this trade-o˛, plotted as an

indi˜erence

curve, which calibrates the

certainty equivalence

associated with the maximisation of an investment™s

expected

NPV (ENPV).

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105 Portfolio Selection and Risk

So far so good, but once an all-equity company has made an issue of ordinary shares and invested the

proceeds, the question management must then address is what to do with the returns. ˜e crucial issue

is whether the ˚rm can increase its total value by the manipulation of ˚nancial risk associated with how

new projects are funded, using retentions at the expense of dividends, rather than a further issue of shares.

And this is where Modigliani and Miller (MM) ˚rst contributed to our analysis.

In Part Two

we presented a popular argument for wealth maximisation based on the investor™s

preference for current dividends, using the Gordon Dividend Growth Model (1962), which proved to

be mathematically ˙awed. Using the economic ﬁlaw of one priceﬂ, MM (1961) also criticised Gordon,

explaining why the shareholders™ desired rate of return (and hence the ˚rm™s cut-o˛ rate for investment

and total value) are una˛ected by dividend policy.

An

all-equity

˜rm can justify the retention of earnings to ˜nance future investments, rather than pay a current dividend,

if their marginal return on new projects at least equals the market rate of interest that shareholders could obtain by

using dividends to ˜nance alternative investments of equivalent business risk elsewhere.

MM believed shareholders should support such managerial behaviour. It cannot detract from their

wealth or consumption preferences, if at any point in time, retentions and dividends are perceived as

perfect economic substitutes

. What they lose through dividends foregone (current income) they expect

to receive through increased equity value (future capital gains) generated by internally ˚nanced projects

discounted at their required opportunity rate of return.

According to MM™s

dividend irrelevancy

hypothesis, if investors periodically need to replace a missing

dividend to satisfy their consumption preferences, the solution is simple.

Ł Shareholders can create a

home-made

dividend by either borrowing an equivalent amount at

the same rate as the company, or sell shares at a price that re˙ects their earnings and reap the

capital gain.

Since the borrowing (discount) rate is entirely determined by the

business

risk of investment (variability

of future earnings) and not the

˝nancial

risk (pattern of dividends), the ˚rm™s distribution policy is trivial.

Ł Dividend decisions are concerned with what is done with earnings but do not determine the

risk originally associated with the quality of investment that produces them.

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106 Portfolio Selection and Risk

In Part ˜ree

we extended our analysis beyond an

all-equity

˚rm to introduce cheaper borrowing (debt)

into the corporate ˚nancial mix. Although the principles of investment using ﬁQuantsﬂ still applied, we

observed that managerial policies designed to maximise shareholder wealth now extend beyond satisfying

shareholder expectations to other stakeholders and comprise the following

inter-related

functions.

Ł ˜e

investment function

, which identi˚es and selects investment opportunities that

maximise

expected

net cash in˛ows

(ENPV) commensurate with risk.

Ł ˜e ˝nance function

, which identi˚es potential fund sources (equity and debt, long or short)

required to sustain investments.

Management therefore need to evaluate the

risk-adjusted

return for each mode of ˚nancing (not just

equity) and. then select an

optimum

mix that will

minimise

the company™s overall weighted average cost

of capital (WACC) as a discount rate for project appraisal.

We then examined the case for and against, an optimum capital structure, which maximises total corporate

value in the presence of gearing (leverage). It became apparent that if a company™s current WACC is to

be employed as a discount rate in ENPV calculations it must satisfy two conditions:

Ł ˜e earnings of new projects selected by management must conform to the pattern (business

risk) of existing corporate activities.

Ł ˜e company™s mode of ˚nance must conform to the composition (˚nancial risk) of its existing

capital structure.

If not, the impact of leverage upon the company™s division of investor returns (overall cost of capital)

and value must also be considered.

Assuming business risk is held constant, the question therefore arose as to whether management can

determine an optimum capital structure that minimises ˚nancial risk and hence the cost of capital as

it gears up with cheaper debt ˚nance. If so, ultimately the ˚rm™s overall value and share price can be

maximised by manipulating ˚nancial policy.

Based on this assumption, what is now termed the ﬁtraditionalﬂ approach, models an

optimum

capital

structure (˚nancial mix) that will

minimise

a ˚rm™s overall weighted average cost of capital (WACC) as

a discount rate for project appraisal and

maximise

total corporate value. However, according to MM™s

cost of capital hypothesis (1958) this strategy can only be a temporary phenomenon.

In this newly leveraged situation, where the ˚nancial returns to debt and equity result from a common

investment decision, MM invoke the ﬁlaw of one priceﬂ to prove that using debt capital to ˚nance new

investment (just like retentions at the expense of dividends) does not matter.

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107 Portfolio Selection and Risk

If two companies in the same class of business risk exhibit a di˛erent WACC and overall value,

arbitrageurs

will begin

trading their investments in the higher valued (geared) company to purchase a cheaper but otherwise perfect substitute

in the lower valued ˜rm (all-equity say) thereby depressing the price of the former and increasing the price of the latter,

until the two are in equilibrium.

As a consequence of

arbitrage

:Ł ˜e total value of a ˚rm represented by the NPV of an income stream discounted at a rate

appropriate to its business risk, should be una˛ected by shi˝s in capital structure (˚nancial risk).

Ł Any rational debt-equity ratio should produce the same cut-o˛ rate for investment (WACC).

In the presence of

arbitrage

, WACC and total corporate value are una˛ected by the debt-equity ratio

(˚nancial risk) but the rather the quality of earnings (business risk) that stems from initial investment.

Today there is considerable empirical support for the MM hypotheses. However, when their papers on

capital structure and dividend policy were ˚rst published in 1958 and 1961 they created controversy

because no theory had been fully developed to explain the pricing of total risk and the relative impact

of its components (business risk and ˚nancial risk) on a

diverse

portfolio of investments. ˜is had to

wait until the publication of the Capital Asset Pricing Model (CAPM) by William Sharpe (1963) based

on Harry Markowitz™ pioneering work on portfolio selection (1952) and the subsequent development

of Modern Portfolio ˜eory (MPT).

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108 Portfolio Selection and Risk

Part Four

of our text therefore considers the evolution of MPT and the contribution of MM™s ﬁlaw of one

priceﬂ to the CAPM and its practical applications. However, what follows assumes that you are familiar

with the body of work by Markowitz and Sharpe and others. If not, you should refer to my

bookboon

texts at the end of this Chapter. If so, here is summary of what they contain as a reminder, with the

maths kept to a minimum!

8.1 Modern Portfolio Theory and Markowitz

Using the following rational investment decision rules based on optimum mean-variance eˆciency

criteria (Quants):

Maximise expected return for a given level of risk

Minimise risk for a given expected return

˜e statistical objective of eˆcient portfolio diversi˚cation is to achieve an overall

standard deviation

, which is lower than its component parts without compromising overall return.

For example, suppose there is a

perfect positive correlation

between two securities that comprise the

stock market, or two projects that de˚ne a ˚rm™s total investment. In other words, high and low returns

always move sympathetically. It would pay an investor, or a company, to place all their funds in whichever

investment yields the highest return at the time. However, if there is

perfect inverse correlation

, where

high returns on one investment are always associated with low returns on the other and

vice versa

, or

there is

random (zero) correlation

between the returns, then it can be shown statistically that overall risk

reduction can be achieved through diversi˚cation.

According to Markowitz (

op.cit

.) if the

correlation coe˚cient

between any number of investments is

less then one (perfect positive), the total risk of a portfolio measured by its standard deviation is lower

than the weighted average of its constituent parts, with the greatest reduction reserved for a correlation

coeˆcient of minus one (perfect inverse).

˜us, if the standard deviation of an individual investment is higher than that for a portfolio in which

it is held, it would appear that some of the standard deviation must have been diversi˚ed away through

correlation with other portfolio constituents, leaving a residual risk component correlated with the

economy as a whole. Measured by the

covariances

of each investment with the total portfolio (such as

the stock market) the latter is undiversi˚able. Consequently, the contribution of an individual investment

to the variance of a well-diversi˚ed portfolio (its covariance) is the only risk that investors will pay a

premium to avoid.

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109 Portfolio Selection and Risk

Indeed, as we shall discover later, the reduction in

total

risk only relates to the

speci˝c

risk associated with

micro-economic

factors, which are unique to individual sectors, companies, or projects. A proportion of

total

risk, termed

market

risk, based on

macro-economic

factors correlated with the market is inescapable.

˜e distinguishing features of speci˚c and market risk had important consequences for the development

of Markowitz eˆciency and the emergence of Modern portfolio ˜eory (MPT) during the 1960s. For

the moment, suˆce it to say that whilst market risk is not diversi˚able, speci˚c risk can be statistically

eliminated entirely if all rational investors diversify until they hold the

market portfolio

, which re˙ects

the risk-return characteristics for every available ˚nancial security. In practice, this strategy is obviously

unrealistic. But as we shall discover later, studies have shown that with less than thirty diversi˚ed

constituents it is feasible to reach a position where a portfolio™s standard deviation is close to that for

the market portfolio.

Unfortunately, throughout the1950s (without today™s computer technology and sophisticated so˝ware)

the derivation of the

covariance

terms in the Markowitz model was so unwieldy for most investors

seeking a well-diversi˚ed risky portfolio drawn from a global capital market that it was untenable. Even

by substituting the

correlation coe˚cient

into the covariance of the portfolio variance, the mathematical

complexity of the variance-covariance matrix calculations for a risky multi-asset portfolio still limited

its applicability. So, what was the alternative?

8.2 Modern Portfolio Theory and the Beta Factor

In an ideal world:

Ł Portfolio theory should o˛er management a practical tool for measuring the extent to which

the pattern of returns from a new project a˛ects the risk of a ˚rm™s existing operations.

Ł For those playing the stock market, portfolio analysis should also calibrate the e˛ects of adding

new securities to their existing spread.

To circumvent the complexity of the Markowitz variance-covariance matrix, various academics sought

alternative ways to measure risk. ˜is began with the realisation that the

total risk

of an investment

(the standard deviation of its returns) within a diversi˚ed portfolio can be divided into

systematic

and

unsystematic

risk. You will recall that the latter (also termed

speci˝c

risk) can be eliminated entirely by

eˆcient diversi˚cation. ˜e other (also termed

market

risk) cannot. It therefore a˛ects the overall risk

of the portfolio in which the investment is included.

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110 Portfolio Selection and Risk

Since all rational investors (including management) interested in wealth maximisation should be

concerned with individual security (or project) risk relative to the stock market as a whole, and not

simply their own asset portfolio, analysts were quick to appreciate the importance of systematic (market)

risk. According to the Separation ˜eorem of John Tobin (1958) it represents the only risk that they will

pay a premium to avoid.

By introducing market opportunities for risk-free investment and borrowing or lending at the risk-free

rate to establish an optimum portfolio, Tobin de˚ned the investor™s required return on a risky investment

as the risk-free return, plus a premium for risk (determined not by the total risk of the investment, but

only by its systematic (market) risk).

Of course, the systematic risk of an individual ˚nancial security (a company™s share, say) might be higher

or lower than the overall risk of the market within which it is listed. Likewise, the systematic risk for some

capital projects may di˛er from others within an individual company™s portfolio. And this is where the

theoretical development of a

relative

measure of an investments systematic risk ˚ts into portfolio analysis.

Termed the

beta

factor (or beta coeˆcient) it calibrates the volatility of say a share™s performance to

market movements (rather than individual securities) de˚ned by the ratio of the expected change in the

stock™s performance to the market itself.

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111 Portfolio Selection and Risk

Expressed statistically:

Ł ˜e beta factor equals the

covariance

of the

jth share™s rate of return with the market portfolio

divided by the

variance

of the market portfolio.

1) ˜j = COV(j,m) VAR(m) For those searching for a computationally simple proxy for the covariance (relative risk) terms in a

Markowitz portfolio, beta neatly solved their problem. Instead of generating numerous new covariance

terms, all they required was the covariance of the rate of return on the additional share with the overall

rate of return on the eˆcient market portfolio.

Activity 1

Since the1960s, innumerable empirical studies have shown that beta values are invaluable for portfolio selection. But

do investors know what they mean?

If you are up to speed with MPT, can you interpret beta factors of 1.15, 1.0 and 0?

Investors can tailor a portfolio to their speci˚c risk-return (utility) requirements, aiming to hold

aggressive

stocks with beta factors in excess of 1.0 while the market is rising (a ﬁbullﬂ market), and less than 1.0

(defensive

) when the market is falling (a ﬁbearﬂ market).

A beta of 1.15

implies that if the underlying market with a beta factor of one were to rise by 10% then

the stock may be expected to rise by 11.5%. Conversely, a security with a beta of less than one would

not be as responsive to market movements. In this situation, smaller systematic risk would mean that

investors would be satis˚ed with a return that is below the market average.

˙e market portfolio

has a beta of one

precisely because the covariance of the market portfolio with itself

is identical to the variance of the market portfolio.

A risk-free investment

has a beta of zero

because its covariance with the market is zero.

8.3 Modern Portfolio Theory and the CAPM

By the 1960s academic research revealed that although a

linear

relationship between total portfolio risk

and expected returns does not hold for individual risky investments using the Markowitz model, all the

characteristics of systematic beta risk apply to portfolios

and

individual securities. ˜e beta of an overall

portfolio is simply the

weighted average

of all it beta factor constituents.

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112 Portfolio Selection and Risk

˜is opened the door to comprehensively pricing market risk and the key for developing the Capital

Asset Pricing Model (CAPM), notably Sharpe™s single period, single index model (

op.cit

). For a given

level of systematic risk, the CAPM determines the equilibrium, mean rate of return for any investment

using its beta value.

Ł ˜e expected return is equal to the risk-free rate of interest,

plus

the product of a market risk

premium (measured by the di˛erence between the market return and the risk free rate) and

the investment™s beta coeˆcient.

For any risky investment (j) with a beta of ˘

j, the expected return (r

j) which provides adequate

compensation for holding the investment is the value obtained by incorporating the beta factor into the

CAPM equation

:2) rj = rf + (rm Œ rf) ˘jWhere r

f and r

m equal the risk-free rate and market return respectively.

And because all the characteristics of systematic betas apply to a

portfolio

, as well as an

individual

security,

any portfolio™s return (r

p) with a portfolio beta (˘

p) can be de˚ned as

3) rp = rf + (rm Œ rf) ˘pConsequently, proponents of the CAPM concluded that all investors are capable of

eliminating unsystematic

risk entirely

by expanding their investment portfolios until they re˙ect their market portfolio (such as

the Dow Jones or Footsie).

Academics also realised the CAPM™s utility, not only within a global stock market framework but also its

relevance to corporate capital budgeting decisions, where an individual project™s beta does not necessarily

equal a ˚rm™s equity beta, let alone the risk measured by its WACC.

Perhaps you recall from Part ˜ree that new projects with di˛erent risk-return trade-o˛s may not conform

to the overall WACC valuation pro˚le of a company. ˜e latter re˙ects no more than the total

average

risk of all its existing investments, which may not even satisfy the aspirations of existing stakeholders,

let alone potential investors. So, if a ˚rm™s WACC is only a

benchmark

, management need to evaluate

the risk of any new asset investment, relative to its existing activities (which may already be diversi˚ed)

as well as the performance of other companies in the same class of business risk

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113 Portfolio Selection and Risk

And this is where we shall pick up on our earlier discussion of MM™s hypotheses and the CAPM in the

following Chapter, by analysing the impact of gearing ratios on beta values (measured by their equity,

asset and project coeˆcients respectively) and the possibility of eliminating their e˛ect on the total

value of a company.

Review Activity

The objective of portfolio diversi˜cation is the selection of investment opportunities that reduce

total

portfolio risk

without compromising

overall

return.

If the standard deviation (risk) of an individual investment is higher than that of the portfolio in which it is held, then

part of the standard deviation must have been diversi˜ed away through correlation with other portfolio constituents.

A high level of diversi˜cation results in rational investors holding the market portfolio, which they will do in combination

with lending or borrowing at the risk-free rate. This only leaves an element of risk that is correlated with the market as

a whole. In other words portfolio risk equals market risk, which is undiversi˜able

To clarify these points for future reference, research and summarise the relationship between

total

risk and its

component

parts.

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114 Portfolio Selection and Risk

An Indicative Outline Solution

Total risk

is divisible between:

-Systematic

or market risk

, so called because it is endemic throughout the system (market)

and is undiversi˚able. It relates to general economic factors that a˛ect all ˚rms and ˚nancial

securities, and explains why share prices tend to move in sympathy.

-Unsystematic risk,

sometimes termed

speci˝c

, residual

, or

unique risk

, relates to speci˚c (unique)

economic factors, which impact upon individual industries, companies, securities and projects.

It can be eliminated entirely through eˆcient diversi˚cation.

In terms of our earlier analysis, systematic risk measures the extent to which an investment™s return moves

sympathetically (systematically) with all the ˚nancial securities that comprise the market portfolio (the

system

). It describes a particular portfolio™s inherent sensitivity to global political and macro-economic

volatility. ˜e best recent example, of course, is the 2007 ˚nancial meltdown and subsequent economic

recession. Because individual companies or investors have no control over such events, they require a

rate of return commensurate with their relative systematic risk. ˜e greater this risk, the higher the rate

of return required by those with widely diversi˚ed portfolios that re˙ect movements in the market as

a whole.

In contrast, unsystematic risk relates to the individual investment and is independent of market risk.

Applied to a company, it is caused by micro-economic factors such as pro˚tability, product innovation

and the quality of management. Because it is completely diversi˚able (variations in returns cancel out

over time) unsystematic risk carries no market premium. ˜us, all the risk in a fully diversi˚ed portfolio

is market or systematic risk.

We have encountered systematic risk earlier in this study under other names. Figure 8.1 reveals that

systemati

c risk comprises a company™s

business risk

and

˝nancial risk.

You will recall that business

risk re˙ects the unavoidable variability of project returns de˚ned by the nature of a ˚rm™s investment

(investment policy

). ˜is may be higher or lower than that for other projects, or the market as a whole.

Systematic risk may also re˙ect a premium for ˚nancial risk, which arises from the proportion of debt

to equity in a ˚rm™s capital structure (gearing) and the amount of dividends paid in relation to the level

of retained earnings, (

˝nancial policy

). Of course, there is empirical support for a contrary view that ˚nancial risk is irrelevant based on the

seminal work of MM (1958 and 1961) explained earlier. Irrespective of whether ˚nancial policies matter,

for the moment all we need say is that for all-equity ˚rms with full dividend distribution policies, there

is an academic consensus that business risk equals systematic (market) risk and is not diversi˚able.

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115 Portfolio Selection and Risk

TOTAL RISK Systematic (Market) Risk Unsystematic (Unique) Risk Business Risk Financial Risk Investment Financial Policy Policy

UNDIVERSIFIABLE DIVERSIFIABLE Figure 8.1:

The Inter-relationship between Risk Concepts

8.4 Summary and Conclusions

At the beginning of this study we outlined theoretically how rational, risk-averse individuals and

companies operating in reasonably eˆcient markets with few ﬁbarriers to tradeﬂ should rank

individual

investments. ˜ey interpret expected returns and standard deviations using the concept of expected

utility to calibrate their risk-return attitudes. In this Chapter we began with the same mean-variance

eˆciency criteria to explain how an optimum

portfolio

of investments can reduce total risk (the standard

deviation) without impairing overall return.

Markowitz

, explains how individuals or companies can reduce risk but maintain their return by holding

more than one investment, providing their returns are not positively correlated. ˜is implies that all

rational investors should diversify risky investments into an eˆcient portfolio. Unfortunately, as its

constituents rise the model not only becomes statistically unwieldy, but also fails to eliminate risk entirely.

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116 Portfolio Selection and Risk

˙e CAPM

fortunately o˛ers investors a statistical lifeline, by discriminating between diversi˚able, non-

systematic and non-diversi˚able, systematic risk. ˜e latter is de˚ned by a beta factor that measures relative

(systematic) risk, which explains how rational investors with di˛erent utility (risk-return) requirements

can choose an optimum portfolio by borrowing or lending at the risk-free rate. Consequently, they are

capable of completely eliminating unsystematic risk by expanding their portfolios until they re˙ect the

market portfolio.

By way of conclusion, however, it is worth noting that without the research expertise and ˚nancial

resources of a global ˚nancial institution required to achieve such extreme diversi˚cation, all is not lost

for private investors with modest funds.

An o˝-forgotten fact (based on numerous studies) is that up to

95 per cent

of unsystematic risk can be

diversi˚ed away by randomly increasing the number of investments in a portfolio to

about thirty

. With

one investment, portfolio risk is represented by the sum of unsystematic and systematic risk. In other

words, the investment™s

total risk

as measured by its standard deviation. When the portfolio constituents

reach double ˚gures, increasingly all the risk associated with holding that portfolio becomes systematic

or market risk.

See Fisher and Lorie (1970) for the earliest and best review of this phenomenon, which is graphed in

Figure 8.2.

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117 Portfolio Selection and Risk

Total Risk Unsystematic Systematic 10 20 30 Number of Shares Figure 8.2:

Portfolio Risk and Diversi˜cation

8.5 Selected References

1. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

2. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

3. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,

Journal

of Business of the University of Chicago,

Vol. 34, No. 4, October 1961.

4. Sharpe, W., ﬁA Simpli˚ed Model for Portfolio Analysisﬂ,

Management Science,

Vol. 9, No. 2,

January 1963.

5. Markowitz, H.M., ﬁPortfolio Selectionﬂ,

˙e Journal of Finance,

Vol.13, No. 1, March 1952.

6. Tobin, J., ﬁLiquidity Preferences as Behaviour Towards Riskﬂ,

Review of Economic Studies

, February 1958.

7. Fisher, L. and Lorie, J., ﬁSome Studies of Variability of Returns on Investment in Common

Stocksﬂ,

Journal of Business,

April 1970.

8. Hill, R.A.,

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118 Portfolio Selection and Risk

Text Books:

Portfolio ˜eory and Financial Analyses, 2010.

Portfolio ˜eory and Financial Analyses: Exercises, 2010.

Business Texts:

Portfolio ˜eory and Investment Analysis,

2010.˜e Capital Asset Pricing Model, 2010.

Portfolio ˜eory and Investment Analysis, 2

nd Edition,

2014.˜e Capital Asset Pricing Model, 2010, 2

nd Edition, 2014.

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119 MM and the CAPM

9 MM and the CAPM

Introduction

So far, our study of portfolio eˆciency, beta factors and the CAPM has concentrated on the stock market™s

analyses of security prices and expected returns by ˚nancial institutions and private individuals. ˜is is

logical because it re˙ects the rationale behind the chronological development of Modern Portfolio ˜eory

(MPT). But what about the impact of MPT on individual companies and their appraisal of capital projects

upon which all investors absolutely depend? If management wish to maximise shareholder wealth, then

surely a new project™s expected return and systematic risk relative to the company™s existing investment

portfolio and stock market behaviour, like that for any ˚nancial security, is a fundamental consideration.

Given your general knowledge of MPT, in this Chapter we shall explore speci˚c corporate applications

of the Sharpe CAPM by strategic ˚nancial management, namely:

-˜e derivation of a discount rate for the appraisal of capital investment projects on the basis

of their systematic risk.

-How the CAPM can be used to match discount rates to the systematic risk of projects that

di˛er from the current business risk of a ˚rm.

Because the model can be applied to projects ˚nanced by debt as well as equity, we shall then conclude

our analyses by establishing a mathematical connection between the CAPM and the Modigliani-Miller

(MM) theory of capital gearing (1958) based on their ﬁlaw of one priceﬂ.

9.1 Capital Budgeting and the CAPM

As an alternative to calculating a ˚rm™s weighted average cost of capital (WACC) explained in Part

˜ree, the theoretical derivation of a project discount rate using the CAPM and its application to NPV

maximisation is quite straightforward. A risk-adjusted discount rate for the

jth project is simply the

risk-free rate added to the product of the market premium and the

project

beta. Using Chapter Eight™s

earlier notation for the

CAPM equation

:4) rj = rf + (rm Œ rf) ˘j˜e project beta (˘

j) measures the

systematic

risk of a speci˚c project (more of which later). For the

moment, suˆce it to say that in many textbooks the project beta is also termed an

asset

beta denoted by ˘

A.Download free eBooks at bookboon.com

120 MM and the CAPM

Using a mathematical formulation, with which you should be familiar, management can then derive a

project™s expected NPV by subtracting the initial cost of investment (I

0) from its periodic, average net

annual cash ˙ows (C

t) discounted at r

j the risk-adjusted rate (rather than WACC ) over its useful life (n).

5) n ENPV = ˜ Ct / (1+rj) t - I0 t=1 Individual

projects are acceptable if:

ENPV 0Collectively

, if ˚nance is a limiting factor (capital rationing) projects that satisfy this acceptance criterion

can also be ranked for selection according to the size of their ENPV. Given:

ENPVA > ENPVB > – ENPVN we prefer project A

So far, so good; but remember that CAPM project discount rates are still based on a number of

simplifying assumptions. Apart from adhering to the traditional concept of perfect capital markets

(Fisher™s Separation ˜eorem) and mean-variance analysis (Markowitz eˆciency) the Sharpe CAPM is

only a

single-period

model, whereas most projects are

multi-period

. According to the CAPM, all investors face the same set of investment opportunities, have the same expectations about

the future and make decisions within

one time horizon. Any new investment made

now

will be realised

then, next year

(say) and a new decision made.

Given the assumptions of perfect markets characterised by random cash ˙ow distributions, there is no

theoretical objection to using a

single-period

model to generate an NPV discount rate for the evaluation

of a ˚rm™s

multi-period

investment plans. ˜e only constraints are that the risk-free rate of interest, the

average market rate of return and the beta factor associated with a particular investment are

constant

throughout its life.

Unfortunately, in reality the risk-free rate, the market rate and beta are rarely constant. However the

problem is not insoluble, as Fama and French observed. (1992). We just substitute

periodic

risk-adjusted

discount rates (now dated r

j t) for a constant r

j into Equation (5) for each future ﬁstate of the worldﬂ, even

if only one of the variables in Equation (4) changes. It should also be noted that the phenomenon of

multiple discount rates combined with di˛erent economic circumstances is not unique to the CAPM.

It is common throughout NPV analyses, as well as other valuation theories (remember the Gordon

Growth Model?).

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121 MM and the CAPM

On ˚rst acquaintance, it would therefore appear that the application of a CAPM return to capital

budgeting decisions provides corporate ˚nancial management with a practical alternative to the

WACC approach.

A particular weakness of WACC is that it de˚nes a single discount rate applicable to

all

projects, based

on the assumptions that their acceptance doesn™t change the company™s risk or capital structure and is

marginal

to existing activities.

In contrast, the CAPM rate varies from project to project, according to the systematic risk of each

investment proposal. However, the CAPM still poses a number of practical problems that must be

resolved if it is to be applied successfully, notably how to derive an appropriate

project

beta factor and

how to measure the impact of

capital gearing

on its calculation.

9.2 The Estimation of Project Betas

So far, we have only used a

general

beta factor (˘) applicable to the

overall

systemic risk of portfolios,

securities and projects. But now our analysis is becoming more focussed,

precise

notation and de˚nitions

are necessary to

discriminate

between systemic

business

and

˝nancial

risk. Table 9.1 summarises the

beta measures that we shall be using for future reference. It also introduces a number of problems with

their application.

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122 MM and the CAPM

˙ = total

systematic

risk, which relates portfolio, security and project risk to

market

risk.

˙j = the business risk of a speci˜c project (

project

risk) for investment appraisal.

˙E = the published equity beta for a company that incorporates business risk and systematic

˜nancial risk if the ˜rm is

geared.

˙A =

the overall business risk of a ˜rm™s

assets (projects). It also equals a company™s

deleveraged

published beta (˙E) which measures business risk

free

of ˜nancial risk.

˙D = the beta value of debt (which obviously equals zero if it is risk-free).

˙EU and ˙EG are the respective equity betas for

similar all-share and geared companies

Table 9.1:

Beta Factor De˜nitions

When an all-equity company is considering a new project with the same level of risk as its current

portfolio of investments, total systematic risk

equals

business risk, such that:

˘ = ˘j = ˘E = ˘A = ˘EUSo, if a company is funded by a combination of debt and equity, this series of equalities must be modi˚ed

to incorporate a

premium

for systematic

˝nancial

risk. As we shall discover, the equity beta will be a

geared

beta re˙ecting business risk

plus

˚nancial risk, which measures shareholder exposure to debt in

their ˚rm™s capital structure. ˜us, the equity beta of an all-share company is always lower than that for

a geared ˚rm with the same business risk.

˘EU < ˘EGTable 9.1 reveals a further idiosyncrasy of the CAPM. A company™s asset beta (˘

A) represents a discount

rate that is appropriate for evaluating projects with the same overall risk as the company itself. But what

if a new project does not re˙ect the average risk of the company™s assets?

You will recall from Part ˜ree that irrespective of gearing, WACC poses a dilemma for management. It

should only be used as a project discount rate if the risk of new investment equals the opportunity cost

of its existing operations.

So too, with the CAPM:

Ł ˜e company™s asset beta (˘

A) produces a discount rate that is only appropriate for evaluating

projects with the same overall risk as the company itself.

Ł Where a new project does not re˙ect the average risk of the company™s assets, the use of

an asset beta is no more likely to produce a correct investment decision than the use of a

WACC calculation.

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123 MM and the CAPM

To illustrate the point, Figure 9.1 graphs the

Security Market Line

(SML) explained in my

bookboon

Portfolio and CAPM series. ˜is shows the required return on a project for di˛erent beta factors, relative

to a company™s overall cost of capital (WACC). ˜e use of WACC to evaluate projects whose risk di˛ers

from the company™s average will be

sub-optimal

where the Internal Rate of Return (IRR) of a project is

in either of the two shaded sections. To calculate the correct CAPM discount rate using Equation (4)

we must determine the

project

beta.

˜e company™s average beta, shown in the diagram, provides a measure of risk for the ˚rm™s overall

returns compared with that of the

market

. However, management™s investment decision is whether or not

to invest in a

project

. So, like the WACC, if the project involves diversi˚cation away from the ˚rm™s core

activities, we must use a beta coeˆcient appropriate to that class of investment. ˜e situation is similar

to a stock market investor considering whether to purchase the shares of the

company

. ˜e individual

would need to evaluate the share™s return by using the

market

beta in the CAPM.

Project Beta Average Beta of Firm™s Assets rfExpected Return Company Cost of Capital (WACC) Security Market Line Figure 9.1:

The SML, WACC and Project Betas

Even if diversi˚cation is not contemplated, the project™s beta factor may not conform to the

average

for the

˚rm™s assets. For example, the investment proposal may exhibit high

operational gearing (

the proportion

of ˚xed to variable costs) in which case the project™s beta will exceed the average for existing operations.

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124 MM and the CAPM

A serious con˙ict (an

agency

problem) can also arise for those companies producing few products,

or worse still a single product, particularly if management approach their capital budgeting decisions

based on self-interest and short-termism, rather than shareholder preferences. Shareholders with well-

diversi˚ed corporate holdings who dominate such companies may prefer to see projects with high risk

(high beta coeˆcients) to balance their own portfolios. Such a strategy may carry the very real threat of

corporate bankruptcy but in the event may have very little impact on their overall returns. For the ˚rm™s

management, other employees, its suppliers and creditors, however, the policy may be economic suicide.

Fortunately, if a beta is required to validate the CAPM for project appraisal, help is at hand. Management

can obtain factors for companies operating in similar areas to the proposed project by subscribing to the

many commercial services that regularly publish beta coeˆcients for a large number of companies, world

wide. ˜eir listings also include stock exchange classi˚cations for

industry

betas. ˜ese are calculated

by taking the market average for quoted companies in the same industry. Research reveals that the

measurement errors of individual betas cancel out when industry betas are used. Moreover, the larger

the number of comparable beta constituents, the more reliable the industry factor.

So, if management wish to estimate a project™s beta, it can identify the industry in which the project falls,

and use that industry™s beta as the project™s beta. ˜is approach is particularly suitable for companies that

are highly

diversi˝ed

and

divisionalised

because their WACC or market beta would be of little relevance

as a discount rate for its divisional operations.

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125 MM and the CAPM

As an alternative to stock market data, management can also estimate a project™s beta from ˚rst principles

by calculating its

F-value

.The F-value of a project is rather like a beta factor in that it measures the variability of a project™s performance,

relative

to the performance of an entity for which a beta value exists.

The entity could be the industry in which the project falls, the ˜rm undertaking the project, or a division within the ˜rm

that is responsible for the project.

A project™s F-value is de˚ned as follows:

6) F = Percentage change in the project™s performance Percentage change in the ﬁentity™sﬂ performance As a result, we can obtain an estimate of a project™s beta through

one of three routes:

(i) % change in the company™s performance x ˜ industry % change in the industry™s performance (ii) % change in the project™s performance x ˜ company ˜ project % change in the company™s performance (iii) % change in the project™s performance x ˜ division % change in the division™s performance Activity 1

A company™s divisional management is considering a capital project, whose performance may be a˛ected 15 per cent

either way, depending on whether the division™s overall performance rises or falls by 10 per cent. In other words, the

project™s pro˜tability is expected to be more volatile than that of the division because of speci˜c economic factors.

Calculate the project™s F-value and beta coe˝cient, given the division™s beta factor is 0.80.

Using Equation (6) we can calculate the F-value as follows:

F = 15% /10% = 1.5If the divisional beta value is 0.80, then the project beta (˘

project

) can be estimated as follows:

(% change in the project™s performance / % change in the division™s performance) × ˘

division

˘ project

= 1.5 × 0.80 = 1.2Download free eBooks at bookboon.com

126 MM and the CAPM

9.3 Capital Gearing and the Beta Factor

˜e CAPM de˚nes an individual investment™s risk relative to a well-diversi˚ed portfolio as

systematic risk.

Measured by the beta coeˆcient, it is the only risk that a company, or an investor, will pay a premium

to avoid. You will recall from Chapter Eight (Figure 8.1) that systematic risk can be sub-divided into:

Ł Business risk

that arises from the variability of a ˚rm™s earnings caused by market forces,

Ł Financial risk

associated with dividend policies and capital gearing, both of which may amplify

business risk

Without getting enmeshed in the dividend debate (covered in Part Two) if we accept the 1961 Modigliani

and Miller (MM) hypothesis as a benchmark, namely that dividends are

irrelevant

(based on their

economic ﬁlaw of one priceﬂ)

˝nancial

risk should not matter for an all-equity company. Applied to the

CAPM, the

systematic

risk of all investors (who are shareholders) can therefore be de˚ned by the

business

risk of the ˚rm™s underlying asset investments.

˜e

equity

beta of an unlevered (all-equity) ˚rm equals an

asset

beta, which measures the business risk

of its total investment relative to the market for ordinary shares (common stock).

Using earlier notation:

˘E U = ˘A ˜e CAPM return on project (r

j) is then de˚ned by:

7) rj = rf + (rm Œ rf) ˘AIf there is no debt in the ˚rm™s capital structure, the company™s asset (equity) beta equals the

weighted

average

of its individual project betas (˘

i) based on the market value of equity:

8) ˘A = wi ˘i = ˘EU where w

i represents the individual weights.

But what about companies who decide to fund future investments by gearing up, or the vast majority

who already employ debt ˚nance?

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127 MM and the CAPM

To make rational decisions, it would appear that management now require an asset beta, which measures

a ˚rm™s business risk that an ungeared equity beta can no longer provide. For example, an all-equity

company may be considering a take-over that will be ˚nanced entirely by debt. To assess the acquisition™s

viability, management will now need to calculate their overall CAPM return on investment, using an

asset beta that re˙ects a

leveraged

˚nancial mix of ˚xed interest on debt and dividends on shares.

Later in this Chapter we shall resolve the dilemma, using the predictions of MM™s capital structure

hypothesis (

op.cit.

). Based on their ﬁlaw of one priceﬂ, whereby similar ˚rms with the same risk

characteristics (except capital gearing) cannot sell at di˛erent prices, it con˚rms their dividend hypothesis,

namely that ˚nancial policy is irrelevant. First, however, let us develop the CAPM, to illustrate the

relationship between an asset beta and the equity and debt beta coeˆcients for a geared company.

You perhaps recall from Part ˜ree that when a ˚rm is ˚nanced by a debt-equity mix, its earnings

stream and associated risk is divided between the ˚rm™s shareholders and providers of corporate debt.

˜e proportion of risk re˙ects the market values of debt and equity respectively, de˚ned by the

debt-

equity ratio

. So, the equity beta will be a

geared

equity beta. It not only incorporates business risk. It

also determines shareholders™ exposure to ˚nancial risk de˚ned by the proportion of contractual, ˚xed

interest securities in the capital structure.

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128 MM and the CAPM

For this reason, as mentioned earlier, the equity beta of an unlevered company is always lower than the

beta of a levered company.

˘EU < ˘EGGiven a geared equity beta (˘

EG) and debt beta (˘

D), the asset beta (˘

A) for a company™s investment in

risky capital projects can be expressed as a weighted average of the two:

9) ˘A = ˘E G [VE / (VE + VD)] + ˘D [VD / (VE + VD)] Where:

VE and V

D de˚ne the

market

values of equity and debt, respectively,

VE plus V

D de˚ne the ˚rm™s total market value (V).

Activity 2

A ˜rm with respective market values of •60m and •30m for equity and debt has an equity beta of 1.5. The debt beta is zero.

Use Equation (9) to calculate the asset beta (˙

A) and explain its mathematical structure.

Ł ˜e asset beta (˘

A) calculation

9) ˘A = ˘E G [VE / (VE + VD)] + ˘D [VD /(VE + VD)]= 1.5 [60/(60 +30)] + 0 [30/(60 +30)] = 1.0Ł ˜e mathematical structure of ˘

AWhen a company is ˚nanced by debt and equity, management need to derive an asset beta using the

weighted average

of its geared equity and debt components. ˜e market values of debt and equity provide

the weightings for the calculation. Note, however, that because the market risk of debt (˘

D) was set to

zero, the right hand side of Equation (9) disappears.

˜is is not unusual. As explained in Part ˜ree, debt has priority over equity™s share of pro˚ts and the sale

of assets in the event of liquidation. ˜us, debt is more secure and if it is risk-free, there is no variance.

So, if ˘

D equals zero, our previous equation for an asset beta reduces to:

10) ˘A = ˘EG [VE / (VE + VD)]

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129 MM and the CAPM

For example, if a company has an equity beta of 1.20, a debt-equity ratio of 40 per cent and we assume

that debt is risk-free, the asset beta is given by:

˘A = 1.20 [100 / (100 + 40)] = 0.86 Perhaps you also recall that debt is a

tax deductible

expense in many economies. Incorporating this

˚scal adjustment into the previous equations (where t is the tax rate) we can rede˚ne the mathematical

relationship between the asset beta and its geared equity and debt counterparts as follows.

11) ˘A = ˘E G {VE / [VE + VD(1-t)]} + ˘D {[VD(1-t) / (VE + VD(1-t))]}

12) ˘A = ˘E G {VE / [VE + VD(1-t)]} if debt is risk-free

Despite the tax e˛ect, our methodology for deriving a company™s asset beta still reveals a

universal

feature

of the CAPM that ˚nancial management can usefully adopt to assess individual projects.

Ł Whenever risky investments are combined, the asset beta of the resultant portfolio is a

weighted

average

of its component parts.

Activity 3

Consider a company with a current asset beta of 0.90. It accepts a project with an asset beta of 0.5 that is equivalent to

10 per cent of its corporate value after acceptance.

Con˜rm that:

1. The

new (ex-post

) asset beta coe˝cient of the company equals 0.86.

2. The

new project reduces the original risk of the ˜rm™s

existing portfolio.

Ł ˜e

ex-post

asset beta coeˆcient

A˝er the project™s acceptance the beta factor equals a weighted average of the ﬁold and newﬂ

de˚ned as:

˘A = (0.90 × 0.9) + (0.5 × 0.1)= 0.86 Ł ˜e revised portfolio

˜e signi˚cance of the project™s acceptance is that with an asset beta of 0.86 compared to 0.90,

the ˚rm™s overall portfolio of investments is now less risky than it was.

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130 MM and the CAPM

9.4 Capital Gearing and the CAPM

For a given level of systematic risk, we know that the CAPM determines the mean rate of return for any

investment,

via

its beta value.

Ł ˜e required return is equal to the risk-free rate of interest, plus the product of the market

premium and the investment™s beta coeˆcient.

From a capital market perspective, for example, the required return on equity that provides adequate

compensation for holding shares is the value obtained by substituting the appropriate equity beta into

the CAPM. From a managerial viewpoint, we therefore have an important policy prescription:

Ł ˜e

shareholders™

equilibrium rate of return, given by the basic CAPM, must equal the

company™s

cost of equity capital.

Turning to individual companies, the CAPM also de˚nes a project™s discount rate as a return equal to

the risk-free rate of interest, plus the product of the market premium and the project™s asset beta (a risk

premium) to compensate for systematic (business) risk. However, we now know that the ˚nancial risk

associated with capital gearing can a˛ect beta factors. So, the discount rate derived from the CAPM for

investment appraisal must also be a˛ected; but how?

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131 MM and the CAPM

Let us ˚rst consider a company funded entirely by equity that is evaluating a new project with the same

level of risk as its existing activities. ˜e ˚rm™s unlevered equity beta (˘

EU) can be used as the project™s

asset beta (˘

A) because the shareholders™ unlevered return (K

eU) equals the company™s return (r

j) on a

new project of equivalent risk. So, the project return that provides adequate compensation for holding

shares in the company is the equity return (K

eU) obtained by substituting the appropriate equity beta

(˘EU) into the familiar CAPM formula.

13) KeU = rj = rf + (rm Œ rf) ˘EU˜e CAPM therefore o˛ers management an important alternative to the derivation of project discount

rates that use the traditional dividend or earnings share valuation models explained earlier in our text.

For an

unlevered

(all-equity) ˚rm, the

shareholders™

return (K

eU) de˚nes the

company™s

cost of capital

(KU) as follows:

14) KU = KeU = rj = rf + (rm Œ rf) ˘EU˜e question we must now ask is whether Equation (14) has any parallel if the ˚rm is geared?

Ignoring ˚scal policy, the short answer is yes. You will recall from Part ˜ree, that in the long run, overall

corporate returns are distributed between shareholders and debt-holders, which represent the cost of

satisfying each capital provider. If no tax bene˚t is conferred on the company through the acquisition

of debt, we can therefore rede˚ne this overall cost by using the CAPM. ˜e ˚rm™s levered WACC (K

G) which obviously varies as the value of debt and equity (V

D and V

E) moves with the market, will equal

the return on the company™s assets in equilibrium.

Rather than use traditional dividend, earnings and interest valuation models to derive a managerial

WACC explained in Part ˜ree, we can substitute an appropriately

geared

asset beta for an

all-equity

beta

into the CAPM to estimate the overall return on debt and equity capital for project appraisal as follows:

15) KG = rj = rf + (rm Œ rf) ˘AOf course, this relationship between WACC and the CAPM only applies in equilibrium when equity and

debt are both fairly priced and the tax system is

neutral

. So, what happens when the system is biased in

favour of the tax deductibility of debt, which ˚gured so prominently throughout Part ˜ree?

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132 MM and the CAPM

In the presence of taxation, there is no connection between the CAPM™s required return on assets and a

company™s WACC. ˜e former is independent of the ˚nancing of assets, whereas the latter is distorted

by the ˚scal relief that interest brings. If the e˛ect of debt ˚nancing is to be eliminated, we must discover

a company identical in every respect to our own, but without any gearing in its capital structure. Such a

company should have the same asset beta factor, since its business risk (the variability of asset returns)

is identical. However, because there is no debt, the ˚rm™s asset beta and the equity beta will exhibit the

same values. ˜us, we can conclude that:

Ł ˜e asset beta for any company, irrespective of its ˚nancial policy (with or without tax) equals

the equity beta of an ungeared company in the same class of business risk.

9.5 Modigliani-Miller and the CAPM

˜is discussion of companies within the same risk class reminds us yet again of MM™s ﬁlaw of one priceﬂ

(op.cit

.). But isn™t this logical?

Ł ˜ere is no theoretical objection to combining MM™s dividends and capital structure hypotheses

within the CAPM. Both their models are entirely consistent with one another, whilst the

assumptions that underpin MM and Modern Portfolio ˜eory (MPT) also stem from a common

source, namely the Separation ˜eorem of Fisher (1930) where:

-Investors are rational and risk averse.

-Investors face the same opportunity set of investments, have the same expectations about

the future and make one period decisions.

-Investors measure risk by the standard deviation of expected returns.

-Information concerning the mean-variance characteristics of investments is freely available.

-˜ere are no transaction costs.

-˜e tax system is neutral.

So, let us conclude our analysis with

equilibrium

formulae for the relationship between the equity betas

of companies in the same risk class (whose asset betas are obviously identical) by comparing the MM

cost of capital hypothesis (WACC) with the CAPM.

Without debt in it capital structure, a company™s asset beta equals its equity beta for projects of equivalent

risk. However, according to MM™s capital structure theory and their

arbitrage

process (explained in

Chapter Seven) companies that are identical in every respect apart from their gearing should also have

the same asset betas. Because their business risk is the same, the factors are not in˙uenced by methods

of ˚nancing.

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133 MM and the CAPM

To summarise MM™s CAPM position, which is entirely consistent with their cost of capital hypothesis

and the derivation of WACC:

Ł An ungeared company™s asset beta equals its equity beta.

Ł A geared company™s asset beta is lower than its equity beta.

Ł Irrespective of gearing, the asset beta for any company equals the equity beta of an ungeared

company with the same business risk.

Ł ˜e asset beta (equity beta) of an unlevered company can be used to evaluate projects in the

same risk class without considering their ˚nance.

˘j = ˘A = ˘EU < ˘E G You will recall from previous Chapters that MM™s capital theory (like their dividend irrelevancy

hypothesis) depends on perfect market assumptions. However, because these assumptions also underpin

much else in ˚nance (including the CAPM) we shall accept them to illustrate the MM relationship

between the beta factors of all-equity and geared companies with the same systemic business risk.

Let us begin with the following CAPM equation, based on Equation (9), in a taxless world.

16) ˘A = ˘E U = ˘E G [VE / (VE + VD)] + ˘D [VD / (VE + VD)]

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134 MM and the CAPM

If we now rearrange terms, divide through by V

E and solve for ˘

EG, the mathematical relationship between

the geared and ungeared equity betas can be expressed as follows:

17) ˘EG = ˘E U + (˘EU Œ ˘D). VD / VE ˜is equation reveals that the equity beta in a geared company equals the equity beta for an all-share

company in the same class of business risk,

plus

a premium for systemic ˚nancial risk. ˜e premium

represents the di˛erence between the all-equity beta and debt beta multiplied by the debt-equity ratio.

However, the important point is that the

increase

in the equity beta measured by the risk premium is

exactly

o˜set

by a lower debt factor as the ˚rm gears up, leaving the asset beta una˛ected. In other words,

irrespective of leverage, the asset betas of the two ˚rms are still identical and equal the equity beta of

the ungeared ˚rm.

˘A = ˘EU < ˘E G For those of you familiar with MM™s capital structure hypothesis outlined in Chapter Seven, the parallels

are striking.

According to MM, the expected return on equity for a geared ˚rm (K

eG) relative to the return (K

eU) for

an all-share ˚rm in a taxless world equals:

18) KeG = KeU + (KeU Œ Kd). VD / VE.˜is states that the return for a geared ˚rm equals an all-equity return for the same class of business risk,

plus

a ˚nancial risk premium de˚ned by the di˛erence between the all-equity return and the cost of debt

multiplied by the debt-equity ratio. ˜e premium compensates shareholders for increasing exposure to

˚nancial risk as a ˚rm gears up. And as we also observed in Part ˜ree, because the cheaper cost of debt

exactly o˛sets rising equity yields, the overall cost of capital (WACC) is una˛ected. So, irrespective of

leverage, all ˚rms with the same business risk can use the cost of equity for an all-share ˚rm as a project

discount rate before considering methods of ˚nancing.

Turning to a world of taxation, where debt is a

tax-deductible

expense with a tax rate (t), we can rede˚ne

the expected return on equity for a geared ˚rm (K

eG) relative to the return (K

eU) for an all-share ˚rm

using a WACC formulation as follows:

19) KeG = KeU + (KeU Œ Kd) (1-t). VD / VE.Likewise, using Equation (17), the equity beta of a geared company is de˚ned by:

20) ˘EG = ˘E U + [(˘EU Œ ˘D) (1-t). VD / VE ] Download free eBooks at bookboon.com

135 MM and the CAPM

And if debt is risk-free with

zero

variance, so that ˘

D equals zero, the formula simpli˚es to:

21) ˘EG = ˘E U + [(˘EU (1-t). VD / VE ] = ˘EU {1 + [(1-t). (VD/VE)]}Review Activity

To illustrate the union between MM and the CAPM, consider Clapton plc, a leveraged company in an economy where

interest is tax deductible at a 20 per cent corporate rate.

200 million ordinary shares (common stock) are authorised and issued at a current market value of £2.00 each (

ex-div

). The equity beta is 1.5.

Debt capital comprises £100 million, irredeemable 10 per cent loan stock, currently trading at par value and the risk-

free rate.

Required:

1. Calculate the company™s asset beta and brie˚y explain the result.

2. If you are mathematically minded, review the previous relationships between the asset beta, the CAPM and

WACC outlined in this Chapter as a basis for project appraisal.

An Indicative Outline Solution

1. ˜e Beta factor

Since the equity beta for an

ungeared

company equals the asset beta for any company in the same risk

class, we can use Equation (20) or better still (21) to solve for ˘

EU and hence ˘

A as follows.

First, de˚ne the market values of Clapton™s equity and debt

VE = £2.00 × 200 million = £400 million

VD

= £100 million

Next, de˚ne the geared equity beta of 1.5 assuming that debt sold at par is risk-free (˘

D = 0).˘EG = 1.5 = ˘EU + [˘EU (1-0.2) (100/400)] = ˘EU {1 + [(1-0.2) (100/400)]}Finally, rearrange terms to solve for ˘

EU and ˘

A : ˘A = ˘EU = 1.5/1.2 = 1.25Download free eBooks at bookboon.com

136 MM and the CAPM

˜e result is to be expected. ˜e asset beta should be smaller than the geared equity beta (

i.e. 1.25 < 1.5) since the systemic risk associated with the asset investment is only one component of the total risk

associated with the shares. ˜e asset beta measures business risk, whereas the geared beta measures

business and ˚nancial risk

2. ˜e Asset Beta, CAPM and WACC

If management use the CAPM, rather than WACC, to obtain a risk-adjusted discount rate for project

appraisal, ˚rst they need to resolve the sequential questions summarised in Table 9.2.

Question: Is the business risk of a project equivalent to that for the company? Answer: YES NO Solution: Use the company™s current Use an equity beta for similar equity beta companies with similar projects Question: Is the chosen equity beta affected by capital gearing? Answer: YES NO Solution: De-leverage ﬁungearﬂ the Use an equity beta equivalent to an equity beta to derive an asset beta if it is not affected by gearing asset beta Table 9.2:

Derivation of an Asset Beta

Having obtained an appropriate asset beta, the project discount rate may then be calculated using our

previous equations, beginning with the basic CAPM formula:

7) rj = rf + (rm Œ rf) ˘AAccording to MM, the asset betas of companies, or projects, in the same class of business risk are identical

irrespective of leverage. Higher equity betas are o˛set by lower debt betas, just as higher equity yields

o˛set cheaper debt ˚nancing when a ˚rm gears up.

Even in a world where debt interest is tax deductible, it is possible to establish a connection between

MM and the CAPM.

˜e MM cost of equity for a geared ˚rm (WACC) is given by:

19) KeG = KeU + [(KeU Œ Kd) (1-t). VD / VE]Download free eBooks at bookboon.com

137 MM and the CAPM

According to the CAPM, the

shareholders™

return (K

eU) for an

unlevered

(all-equity) ˚rm, de˚nes the

company™s

cost of capital (K

U) as follows:

13) KeU = rj = rf + (rm Œ rf) ˘EUAnd for a

geared

˚rm, the corresponding equity return (K

eG) is given by:

22) KeG = rj = rf + (rm Œ rf) ˘EGWhere: ˘

A = ˘EU < ˘EG If we assume that the company™s pre-tax cost of debt (K

d) in Equation (19) equals the risk-free rate (r

f) in Equations (13) and (22) remember we can rewrite r

f for K

d in Equation (19).

If we then substitute Equations (13) and (22) into Equation (19) rearrange terms and simplify the result,

we can con˚rm our earlier equations for a

geared

equity beta:

20) ˘EG = ˘E U + [(˘EU Œ ˘D) (1-t). VD / VE ] Download free eBooks at bookboon.com

138 MM and the CAPM

And if debt is risk-free with

zero

variance, so that ˘

D equals zero, the formula simpli˚es to:

21) ˘EG = ˘E U + [(˘EU (1-t). VD / VE ] = ˘EU {1 + [(1-t). (VD/VE)]}Practical applications of these equations and the derivation of an equilibrium cost of equity and WACC

using the CAPM are referenced in the companion Exercise Text (the next in my

bookboon

series). It also

contains detailed examples of MM™s comprehensive contribution to modern ˚nance, which support all

the previous Chapters, as a guide to your future studies.

9.6 Summary and Conclusions

˜is entire study is based upon a mean-variance analysis of investment decisions within a framework

of uncertainty, using the normative objective of shareholder wealth maximisation and the assumptions

of a perfect capital market, which we initially accepted without criticism.

Because ownership is divorced from control (the

agency

principal) we then argued that if management

wish to maximise shareholders™ wealth (using equity value as a proxy), companies ought to consider

the consequences of their actions. According to conventional ˚nancial theory, every capital

investment

decision is inextricably tied to a ˚rm™s operational and strategic

˝nancial

decisions, which include:

Ł ˜e expected NPV maximisation of all a ˚rm™s projects (Part One).

Ł ˜e relevance of an optimal dividend policy, rather retained earnings (Part Two).

Ł ˜e determination of an optimal capital structure through the issue of debt, rather than equity

(Part ˜ree).

So, for any given investment policy, the pivotal issue is whether a ˚rm can maximise its total value by

manipulating its ˚nancial policies.

Modigliani and Miller comprehensively rubbished this view nearly sixty years ago. Using

arbitrage

in perfect capital markets, they demonstrated that ˚nancial policy does not matter.

The total market value of any company is independent of its dividend policy and capital structure, and is found by

capitalising expected returns at a discount rate appropriate to its class of business risk.

In Part Four, the analysis of investment returns and the pricing of risk within a portfolio framework

con˚rms their hypotheses. A detailed consideration of MPT, based on Markowitz eˆciency, the beta

coeˆcient in its various guises and the CAPM revealed that:

Ł ˜e value of a levered ˚rm in general equilibrium is equal to its unlevered counterpart.

Ł ˜e sum of the values of debt and equity are based on the returns to each.

Ł ˜e sum of the returns to debt-holders and shareholders must therefore equal the net operating

cash ˙ows of an all-equity ˚rm.

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139 MM and the CAPM

Even within Markowitz™s original frame of reference:

Ł ˜e sum of the covariances of returns to both providers of capital must equal the covariance

of the ˚rm™s net operating cash ˙ows.

Given the normative assumptions of traditional capital market theory, upon which our study is based,

Fisher™s Separation ˜eorem, the MM hypotheses and Modern Portfolio ˜eory (MPT) are theoretically

united. ˜e critical question is whether the relaxation of their common assumptions invalidates their real

world applicability. If so, we must continue the search for more realistic explanations of investor behaviour.

9.7 Selected References

1. Sharpe, W., ﬁA Simpli˚ed Model for Portfolio Analysisﬂ,

Management Science,

Vol. 9, No. 2,

January 1963.

2. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory of

Investmentﬂ,

American Economic Review,

Vol. XLVIII, No. 4, September 1958.

3. Fisher, I.,

˙e ˙eory of Interest

, Macmillan, 1930.

4. Markowitz, H.M., ﬁPortfolio Selectionﬂ,

Journal of Finance

, Vol. 13, No. 1, 1952.

5. Fama, E.F. and French, K.R., ﬁ˜e Cross-Section of Expected Stock Returnsﬂ,

Journal of Finance,

Vol. 47, No. 3, June 1992.

6. Gordon, M.J.,

˙e Investment, Financing and Valuation of a Corporation,

Irwin, 1962.

7. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,

Journal

of Business of the University of Chicago,

Vol. 34, No. 4, October 1961.

8. Hill, R.A.,

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Text Books:

Portfolio ˜eory and Financial Analyses, 2010.

Portfolio ˜eory and Financial Analyses: Exercises, 2010.

Business Texts:

Portfolio ˜eory and Investment Analysis,

2010.˜e Capital Asset Pricing Model, 2010.

Portfolio ˜eory and Investment Analysis, 2

nd Edition,

2014.˜e Capital Asset Pricing Model, 2010, 2

nd Edition, 2014.

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