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7 CAPM and MM combined – geared betas 12/08, 12/10, 6/14

2.7.3 Example: CAPM and geared betas

Two companies are identical in every respect except for their capital structure. Their market values are in

equilibrium, as follows.

Annual profit before interest and tax

Less interest (4,000 8%)

Less tax at 30%

Profit after tax = dividends

Market value of equity

Market value of debt

Total market value of company

Geared

$'000

1,000

320

680

204

476

3,900

4,180

8,080

Ungeared

$'000

1,000

0

1,000

300

700

6,600

0

6,600

The total value of Geared is higher than the total value of Ungeared, which is consistent with MM.

All profits after tax are paid out as dividends, and so there is no dividend growth. The beta value of

Ungeared has been calculated as 1.0. The debt capital of Geared can be regarded as risk free.

Calculate:

(a)

(b)

(c)

The cost of equity in Geared

The market return Rm

The beta value of Geared

Solution

(a)

Since its market value (MV) is in equilibrium, the cost of equity in Geared can be calculated as:

d

476

= 12.20%

MV 3,900

(b)

The beta value of Ungeared is 1.0, which means that the expected returns from Ungeared are

exactly the same as the market returns, and Rm = 700/6,600 = 10.6%.

(c)

e = a

= 1.0

Ve Vd (1 T)

Ve

3,900 (4,180 0.70)

= 1.75

3,900

The beta of Geared, as we should expect, is higher than the beta of Ungeared.

2.7.4 Using the geared and ungeared beta formula to estimate a beta factor

So what is the relevance of geared and ungeared betas?

A private company may want to evaluate a proposed new investment using DCF and so wants to identify a

suitable cost of capital to use as the discount rate. Because it is a private company, it does not have a beta

factor. However, it may identify a listed company that is similar to itself in many ways, and whose beta

factor it can use to establish its own cost of equity. If the listed company and the private company are

similar in every respect except for their gearing, it would be appropriate to adjust the estimated beta factor

for the private company by making an adjustment for the difference in gearing levels between the two

companies.

If a company plans to invest in a project which involves diversification into a new business, the investment

will involve a different level of systematic risk from that applying to the company's existing business. A

discount rate should be calculated which is specific to the project, and which takes account of both the

project's systematic risk and the company's gearing level. The discount rate can be found using the

CAPM.

Part E Business finance 16: Capital structure

327

Step 1

Get an estimate of the systematic risk characteristics of the project's operating cash flows

by obtaining published beta values for companies in the industry into which the company is

planning to diversify.

Step 2

Adjust these beta values to allow for the company's capital gearing level. This adjustment is

done in two stages.

(a)

Convert the beta values of other companies in the industry to ungeared betas, using

the formula:

Ve

a = e

Ve Vd (1 T)

(b)

Having obtained an ungeared beta value a, convert it back to a geared beta e,

which reflects the company's own gearing ratio, using the formula:

e = a

Step 3

Ve Vd (1 T)

Ve

Having estimated a project-specific geared beta, use the CAPM to estimate a project-specific

cost of equity. Having calculated a cost of equity, it may be necessary to calculate a

weighted average cost of capital if there is also debt capital in the financing.

This may seem complicated. An example will be used to illustrate the method.

2.7.5 Gearing and ungearing betas

A company's debt:equity ratio, by market values, is 2:5. The corporate debt, which is assumed to be risk

free, yields 11% before tax. The beta value of the company's equity is currently 1.1. The average returns

on stock market equity are 16%.

The company is now proposing to invest in a project which would involve diversification into a new

industry, and the following information is available about this industry.

(a)

(b)

Average beta coefficient of equity capital = 1.59

Average debt:equity ratio in the industry = 1:2 (by market value)

The rate of corporation tax is 30%. What would be a suitable cost of capital to apply to the project?

Solution

The company should not use its existing WACC as the discount rate for the planned project, because the

investment will be in a different industry or market sector where the systematic risk is different.

Instead it can use the average systematic risk in the 'new' industry to determine a cost of capital. A

problem is that the beta factor for listed companies that are already in the industry is different to some

extent because of the different gearing level.

To get round this problem, we calculate a geared beta for the company based on the average geared betas

of companies already in the industry, adjusted to allow for the difference in gearing.

This is essentially a three-step process.

328

(1)

Convert the geared beta for the new industry into an ungeared beta.

(2)

Use the ungeared beta to calculate a geared beta that reflects the company's own capital structure.

(3)

Use this geared beta to calculate an appropriate cost of equity for the investment. This cost of

equity should be used to determine an appropriate weighted cost of capital to use as the discount

rate.

16: Capital structure Part E Business finance

The beta value for the industry is 1.59.

Step 1

Convert the geared beta value for the industry to an ungeared beta (asset beta) for the

industry.

2

a = 1.59

= 1.18

2 (1(1 0.30))

Step 2

Convert this ungeared industry beta back into a geared beta, which reflects the company's

own gearing level of 2:5.

5 (2 (1 0.30))

e = 1.18

= 1.51

5

Step 3

(a)

This is a project-specific beta for the firm's equity capital and so, using the CAPM,

we can estimate the project-specific cost of equity as:

keg = 11% + (16% – 11%) 1.51 = 18.55%

(b)

The project will presumably be financed in a gearing ratio of 2:5 debt to equity, and

so the project-specific cost of capital ought to be:

[5/7 18.55%] + [2/7 70% 11%] = 15.45%

Question

Ungeared and geared betas

Two companies are identical in every respect except for their capital structure. XY has a debt:equity ratio

of 1:3, and its equity has a value of 1.20. PQ has a debt:equity ratio of 2:3. Corporation tax is at 30%.

Estimate a value for PQ's equity.

Answer

Estimate an ungeared beta from XY data.

a = 1.20

3

= 0.973

3 (1(1 0.30))

Estimate a geared beta for PQ using this ungeared beta.

e = 0.973

3 (2(1– 0.30))

3

= 1.427

2.7.6 Weaknesses in the formula

The problems with using the geared and ungeared beta formula for calculating a firm's equity beta from

data about other firms are as follows.

(a)

It is difficult to identify other firms with identical operating characteristics.

(b)

Estimates of beta values from share price information are not wholly accurate. They are based

on statistical analysis of historical data and, as the previous example shows, estimates using one

firm's data will differ from estimates using another firm's data.

(c)

There may be differences in beta values between firms caused by:

(i)

Different cost structures (eg the ratio of fixed costs to variable costs)

(ii)

Size differences between firms

(iii)

Debt capital not being risk free

Part E Business finance 16: Capital structure

329

(d)

If the firm for which an equity beta is being estimated has opportunities for growth that are

recognised by investors, and which will affect its equity beta, estimates of the equity beta based on

other firms' data will be inaccurate, because the opportunities for growth will not be allowed for.

Perhaps the most significant simplifying assumption is that, to link MM theory to the CAPM, it must be

assumed that the cost of debt is a risk-free rate of return. This could obviously be unrealistic. Companies

may default on interest payments or capital repayments on their loans. It has been estimated that

corporate debt has a beta value of 0.2 or 0.3.

The consequence of making the assumption that debt is risk free is that the formulae tend to overstate the

financial risk in a geared company and to understate the business risk in geared and ungeared companies

by a compensating amount.

Question

Gearing and ungearing betas

Backwoods is a major international company with its head office in the UK, wanting to raise £150 million

to establish a new production plant in the eastern region of Germany. Backwoods evaluates its

investments using NPV, but is not sure what cost of capital to use in the discounting process for this

project evaluation.

The company is also proposing to increase its equity finance in the near future for UK expansion, resulting

overall in little change in the company's market-weighted capital gearing.

The summarised financial data for the company before the expansion are shown below.

STATEMENT OF PROFIT OR LOSS (EXTRACTS) FOR THE YEAR ENDED 31 DECEMBER 20X1

Revenue

Gross profit

Profit after tax

Dividends

Retained earnings

STATEMENT OF FINANCIAL POSITION (EXTRACTS) AS AT 31 DECEMBER 20X1

Non-current assets

Current assets

Total assets

£m

1,984

432

81

37

44

£m

846

350

1,196

Issued ordinary shares of £0.50 each nominal value

Reserves

225

761

986

Medium-term and long-term loans (see note below)

Total equity and liabilities

210

1,196

Note on borrowings

These include £75m 14% fixed rate bonds due to mature in five years' time and redeemable at par. The

current market price of these bonds is £120 and they have an after-tax cost of debt of 9%. Other mediumand long-term loans are floating-rate UK bank loans at LIBOR plus 1%, with an after-tax cost of debt of

7%.

Company rate of tax may be assumed to be at the rate of 30%. The company's ordinary shares are

currently trading at 376p.

The equity beta of Backwoods is estimated to be 1.18. The systematic risk of debt may be assumed to be

zero. The risk-free rate is 7.75% and market return is 14.5%.

The estimated equity beta of the main German competitor in the same industry as the new proposed plant

in the eastern region of Germany is 1.5, and the competitor's capital gearing is 35% equity and 65% debt

by book values, and 60% equity and 40% debt by market values.

330

16: Capital structure Part E Business finance

Required

Estimate the cost of capital that the company should use as the discount rate for its proposed investment

in eastern Germany. State clearly any assumptions that you make.

Answer

The discount rate that should be used is the weighted average cost of capital (WACC), with weightings

based on market values. The cost of capital should take into account the systematic risk of the new

investment, and therefore it will not be appropriate to use the company's existing equity beta. Instead, the

estimated equity beta of the main German competitor in the same industry as the new proposed plant will

be ungeared, and then the capital structure of Backwoods applied to find the WACC to be used for the

discount rate.

Since the systematic risk of debt can be assumed to be zero, the German equity beta can be 'ungeared'

using the following expression.

Ve

a = e V V (1 T)

e

d

where: a

e

Ve

Vd

T

=

=

=

=

=

asset beta

equity beta

proportion of equity in capital structure

proportion of debt in capital structure

tax rate

For the German company:

60

a = 1.5

= 1.023

60

40(1

0.30)

The next step is to calculate the debt and equity of Backwoods based on market values.

Equity

450m shares at 376p

Debt: bank loans

Debt: bonds

Total debt

(210 – 75)

(75 million 1.20)

Total market value

£m

1,692.0

135.0

90.0

225.0

1,917.0

The beta can now be re-geared

e =

1.023(1,692 225 (1 0.3))

= 1.118

1,692

This can now be substituted into the capital asset pricing model (CAPM) to find the cost of equity.

E(r i) = Rf + (E (rm) – Rf)

where: E(r i)

Rf

E(rm)

E(r i)

= cost of equity

= risk-free rate of return

= market rate of return

= 7.75% + (14.5% – 7.75%) 1.118 = 15.30%

The WACC can now be calculated:

1,692

135

90

15.3 1,917 + 7 1,917 + 9 1,917 = 14.4%

Part E Business finance 16: Capital structure

331

Exam focus

point

An exam question may ask you to explain how CAPM can be used in investment appraisal rather than

requiring a calculation.

There is a series of articles on CAPM available on www.accaglobal.com.

332

16: Capital structure Part E Business finance

Chapter Roundup

Some commentators believe that an optimal mix of finance exists at which the company's cost of capital

will be minimised.

Under the traditional theory of cost of capital, the weighted average cost of capital declines initially as

gearing increases, but then rises as gearing increases further. The optimal capital structure is at the

gearing level where WACC is lowest.

Modigliani and Miller stated that, in the absence of tax relief on debt interest, a company's capital

structure would have no impact on its WACC. WACC would be the same regardless of the company's

capital structure.

The lower a company's WACC, the higher the NPV of its future cash flows and the higher its market value.

When an investment has differing business and finance risks from the existing business, geared betas

may be used to obtain an appropriate cost of capital and required rate of return for an investment.

Geared betas are calculated by:

–

–

Ungearing industry betas

Converting ungeared betas back into a geared beta that reflects the company's own gearing ratio

Quick Quiz

1

What are the main problems in using geared and ungeared betas to calculate a firm's equity beta?

2

Explain the significance of lines 1 to 3 and point 4 in the diagram below illustrating the traditional view of

the WACC.

3

Assuming debt is risk free a = ?

4

To use WACC as the discount rate in an investment appraisal, the project must have the same business

risk as the overall company. Why is this?

5

Why, in the real world, do businesses not adopt the Modigliani and Miller (with taxation) theory that a

business should be solely funded by debt?

Part E Business finance 16: Capital structure

333

Answers to Quick Quiz

1

(a)

(b)

(c)

(d)

2

Line 1 is the cost of equity in the geared company.

Line 2 is the weighted average cost of capital.

Line 3 is the cost of debt.

Point 4 is the optimal level of gearing.

3

It is difficult to identify other firms with identical operating characteristics.

Estimates of beta values from share price information are not wholly accurate.

There may be firm-specific causes of differences in beta values.

The market may recognise opportunities for future growth for some firms but not others.

Ve

a = e V V (1 T)

e

d

4

If a new investment has different business risks from the company as a whole then investors may seek a

higher return if they deem the project to be riskier. Conversely, a lower return may be required if the

project is deemed to be less risky.

5

The Modigliani and Miller theory is based on perfect capital markets which do not exist in the real world.

Now try the questions below from the Practice Question Bank

334

Number

Level

Marks

Approximate time

Section A Q28

Examination

2

4 mins

Section C Q19

Examination

20

39 mins

16: Capital structure Part E Business finance

P

A

R

T

F

Business valuations

335

336

Business valuations

Topic list

Syllabus reference

1 The nature and purpose of business valuations

F1 (a), (b)

2 Asset valuation bases

F2 (a)

3 Income-based valuation bases

F2 (b)

4 Cash flow based valuation models

F2 (c)

5 Valuation of debt

F3 (a)

Introduction

In Part F we shall be concentrating on the valuation of businesses. In this

chapter, we will cover the reasons why businesses are valued and the main

methods of valuation.

337

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