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

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

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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
(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.

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(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.

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

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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
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.
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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.

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.

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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

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

335

336

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