ACCA P4 Advanced Financial Management - 2010 - Study text

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Chapter 6: Cost of capital

= $60 million + ($20 million × 0.25) = $75 million.

The market value of the debt capital is $20 million; therefore the market value

of the equity in the geared company is $55 million ($75 million – $20 million).

(b)

WACC of the geared company

The WACC of the company after the increase in its gearing is calculated as

follows:

WACC

= WACC

1−

D + E

()

⎡

⎣

⎢

⎤

⎦

⎥

WACC

= 8% 1−

$20 million × 25%

()

$65 million

()

⎡

⎣

⎢

⎤

⎦

⎥

= 8% 0.9231

()

= 7.38%

(c)

Cost of equity in the geared company

()

KK)1(KK

DEUEUEG

−−+= t

= 8% + [(1 – 0.25) (8 – 5) 20/45] = 8% + 1% = 9%

Check: the WACC can now be calculated as follows:

Source of finance Market value Cost Market value x Cost

$ million r MV x r

Equity 45.00

9.00

405

Debt (after-tax cost) 20.00

3.75

65.00

480

7.38%

65.00

480

WACC ==

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Change in gearing, the WACC and capital investment appraisal

 Modigliani and Miller with taxation: from one level of gearing to another

 The method

 Relevance for capital investment appraisal

6 Change in gearing, the WACC and capital investment

appraisal

6.1 Modigliani and Miller with taxation: from one level of gearing to another

When a company is considering a major new capital investment project (where the

business risk is similar to the risk with the company’s other business operations),

the method of financing the investment might alter the company’s gearing. For

example, if a project is financed entirely by new debt capital, its gearing level will

increase.

 A change in gearing will alter the cost of equity (Modigliani and Miller

proposition 2).

 There might be a change in the cost of debt, where the gearing level rises to such

a high level that ‘financial distress’ concerns make debt capital more expensive.

However, at lower levels of gearing it is assumed that the cost of debt is

unaffected by changes in the gearing level.

 There will be a reduction in the WACC (Modigliani and Miller proposition 1).

If the project is evaluated using the WACC to estimate the NPV, the new WACC

should be used for the NPV evaluation.

This means that when a new capital project will result in a change in gearing, it is

necessary to calculate a new WACC before going on to the NPV calculations.

The Modigliani and Miller formulae can be used to do this. The explanation that

follows concentrates on the formula for the cost of equity, because this is the

formula that you will be given in the examination.

6.2 The method

The approach should be as follows:

 Step 1. Start with the company at its original level of gearing. You should be

given the value of the company (the value of its equity and the value of its debt

capital) and the cost of its equity and debt capital.

 Step 2. Use these values to calculate the value of a comparable ungeared

company, and the cost of equity in the ungeared company. Use the Modigliani

and Miller formulae to do this.

Chapter 6: Cost of capital

 You now have the cost of equity in a comparable ungeared company.

 Step 3. Use these values for the ungeared company to work out values for the

company at its new level of gearing: total value, value of equity, WACC and cost

of equity.

Example

A company has a total current value of $100 million, consisting of $80 million equity

and $20 million of debt capital. The cost of equity is 10% and the pre-tax cost of the

debt capital is 6%. The rate of tax on company profits is 25%.

The company proposes to borrow an additional $20 million of debt capital, and use

the money to buy back and cancel $20 million of its equity.

Required

According to Modigliani and Miller, what will be the following values for the

company at its new level of gearing?

(a) Its total value, divided into a value for the equity and a value for the debt

capital

(b) Its WACC

This is a long example, but you should work through the solution carefully.

Answer

(a) Total value of the company

(i) Step 1: Value of a similar all-equity company.

We have the current value of the geared company, which is $100 million,

consisting of $80 million equity and $20 million debt capital. We can

calculate the cost of a similar company that is all-equity financed.

= V

+Dt

100 million = V

+ (20 million × 0.25)

= 95 million.

(ii) Step 2: Value of the company at the new level of gearing.

The company will be replacing $20 million of equity with $20 million of

debt capital, so in the new gearing structure, debt capital increases.

The market value of the debt will be $20 million + $20 million = $40

million.

We can calculate the total cost of the company at its new gearing level,

using the same MM formula.

= V

+Dt

= 95 million + (40 million × 0.25)

= 105 million.

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The total value of the company at the new gearing level will be $105

million. Of this, $40 million will be debt capital; therefore the value of

the remaining equity will be $65 million.

(b)

WACC

(i) Step 1: WACC of a similar all-equity company.

The WACC of the company at its current level of gearing is calculated as

follows:

Source of finance Market value Cost Market value x Cost

$ million r MV x r

Equity 80.00

0.10

8.00

Debt (after-tax cost)

20.00

0.045

0.90

100.00

8.90

WACC = 8.9/100 = 0.089 or 8.9%.

We can use the MM formula for WACC to calculate what the WACC

would be in a similar ungeared company.

The WACC of a similar ungeared company is:

⎥

⎦

⎤

⎢

⎣

⎡

80)+(20

0.25 20

-1 WACC=0.089

0.089 = 0.95 WACC

WACC

= 0.09368

(ii) Step 2: WACC of the company at the new level of gearing.

Having established the WACC in an all-equity company, we can now

use the same formula to calculate the WACC in the company at its new

level of gearing, with $40 million debt and $65 million of equity.

(The value of equity and debt capital at the new level of gearing were

calculated in (a)).

⎥

⎦

⎤

⎢

⎣

⎡

65)+(40

0.25 40

-1 0.09368=WACC

WACC

= 0.09368 × 0.90476

WACC

= 0.085 = 8.5%.

(c)

Cost of equity

(i) Step 1: Cost of equity of a similar all-equity company.

A similar approach is taken for calculating the cost of equity. We start by

calculating the cost of equity in a similar all-equity company, using the

MM formula for Proposition 2. We know the value of K

and we need

to calculate a value for K

Chapter 6: Cost of capital

In the original geared company, the value of equity is $80 million and

the value of debt capital is $20 million.

The cost of equity in a similar all-equity company is calculated as

follows:

()

KK)1(KK

DEUEUEG

−−+= t

10 = K

+ [(1 – 0.25) (K

– 6) 20/80]

10 = K

+ 0.1875 K

– 1.125

1.1875 K

= 11.125

= 9.3684.

(ii) Step 2: Cost of equity of the company at the new level of gearing.

Having calculated the cost of equity in a similar all-equity company, we

can now calculate the cost of equity in the company at its new level of

gearing. Debt capital is $40 million and equity is $65 million.

(The value of equity and debt capital at the new level of gearing were

calculated in (a).)

()

KK)1(KK

DEUEUEG

−−+= t

= 9.3684 + (1 – 0.25) (9.3684 – 6) 40/65

= 9.3684 + 1.5546 = 10.923, say 10.9%.

6.3 Relevance for capital investment appraisal

The Modigliani and Miler formulae can be used to re-calculate the cost of equity and

the WACC in a company where the level of gearing changes, provided there is no

change in the overall business risk and the company is therefore similar in all

respects except for its gearing.

When a company plans a new capital investment that will alter its gearing, without

affecting its business risk profile, the MM formulae can be used to calculate the cost

of equity and WACC at the new level of gearing. The new WACC can then be used

as the discount rate for calculating the NPV of the proposed project.

If you are given an examination question of this type in your examination, you

might prefer to use the formula for Proposition 2 (cost of equity) because this is

given to you in the formula sheet in the examination, and you do not need to learn

it.

Study the following example carefully.

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Example

A US company has a current total market value of $12 billion, consisting of $10

billion of equity and $2 billion of debt capital. The debt capital will mature in four

years’ time. The current weighted average cost of capital is 6.5%.

The company is considering a new investment costing $3 billion, which it would

finance entirely by $3 billion of new ten-year bonds.

The yield curve for US government bonds (Treasuries) shows that the risk-free cost

of four-year debt is 4% and the risk-free cost of ten-year debt is 4.25%. The credit

rating on the company’s current debt capital is AA, but if the new bond issue takes

place there is a 75% probability that all the company’s debt will be re-rated to AA-

and a 25% probability that all the company’s debt will be re-rated to A+. (This

applies to both the existing debt and the new bonds.)

The spreads for yields on corporate bonds above the US Treasuries yield curve are

as follows:

Credit rating Spreads (basis points)

4-year bonds 10-year bonds

AA 30 50

AA - 40 60

A + 45 70

The rate of taxation on company profits is 30%

Required

Calculate the weighted average cost of capital in the company if the project goes

ahead and is financed entirely by ten-year bonds. Assume that there will be no

change in the company’s business risk.

Answer

Total value of the company

In the previous example, it was possible to re-calculate the total value of the

company using the Modigliani and Miller formula

Dt .

In this example, this is not possible, because the company is raising additional debt

capital. The MM formula applies only when the size of the company is unchanged,

and equity replaces debt or debt replaces equity. The formula cannot be used when

additional debt capital is raised.

An assumption will therefore be made (even though it is unsatisfactory) that when

the company obtains its additional debt capital of $3 billion, the value of the equity

will be unchanged at $10 billion.

Chapter 6: Cost of capital

Preliminary workings

We need to calculate the current cost of equity in the company, in order to use the

MM formula for Proposition 2. We know the current WACC which is 6.5%. We also

know that the current credit rating for the four-year debt capital is AA, which

means that the pre-tax cost of debt is the risk-free yield of 4.00% plus a spread of 30

basis points. The pre-tax cost of debt is therefore 4.30% and the after-tax cost of debt

is 4.30% (1 – 0.30) = 3.01%.

We now have enough data to calculate the

current cost of equity.

The WACC, which is 6.5%, is calculated as follows:

Source of finance Market

value

Cost

Market value

× Cost

$ billion

MV × r

Equity 10.0

10r

Debt 2.0

3.01

6.02

12.0

10r + 6.02

WACC = 6.5% =

6.02 10r +

6.5 (12) = 10r + 6.02

10r = 71.98

r = 7.198%

The current cost of equity is 7.198%.

We also need to calculate the

cost of debt in the company after the new bond issue.

If AA - If A +

4 year 10 year

4 year

10 year

Risk-free 4.00 4.25

4.00

4.25

Spread 0.40 0.60

0.45

0.70

Pre-tax cost

4.40 4.85

4.45

4.95

We can calculate a weighted average pre-tax cost of the debt capital, as follows:

Market value AA - A +

$bn Cost

Cost × MV

Cost

Cost × MV

4 year 2 4.40

8.80

4.45

8.90

10 year 3

4.85

14.55

4.95

14.85

23.35

23.75

Weighted average before-tax cost of debt if credit rating is AA - = 23.35/5 = 4.67%.

Weighted average before-tax cost of debt if credit rating is A + = 23.75/5 = 4.75%.

Expected weighted average before tax cost of debt = (0.75 × 4.67) + (0.25 × 4.75)%

= 4.69%.

Expected weighted average after-tax cost of debt = 4.69% (1 – 0.30) = 3.283%.

Paper P4: Advanced financial management

Calculate the cost of equity in the company after the new bond issue

Step 1. Calculate the cost of equity in an ungeared company at the current level of

gearing. We know that the current cost of equity at the current level of gearing is

7.198%.

()

KK)1(KK

DEUEUEG

−−+= t

7.198 = K

+ [(1 – 0.30) (K

– 4.30) 2/10]

7.198 = K

+ 0.14 K

– 0.602

1.14 K

= 7.8

= 6.842.

Step 2. Calculate the cost of equity in the company at the new level of gearing,

making the assumption about market value discussed above, but

assuming no

change in the cost of debt

. (Changing the cost of debt in this calculation would

produce an incorrect valuation for the cost of equity.)

()

KK)1(KK

DEUEUEG

−−+= t

= 6.842 + 0.890

= 7.732.

Calculate the WACC at the new level of gearing

We can now calculate the WACC after the bond issue.

Source of finance Market

value

Cost

Market value

× Cost

$ billion

MV × r

Equity 10.0

7.732

77.320

Debt (weighted average) 5.0

3.283

16.415

15.0

93.735

WACC = 93.735/15 = 6.249, say 6.25%.

A discount rate of 6.25% should be used to calculate the NPV of the proposed new

capital investment.

Chapter 6: Cost of capital

Asset beta

 Changes in gearing, the cost of equity and equity beta factors

 Definition of asset beta

 The asset beta formula

 Using the asset beta formula

 The asset beta formula: changing from one level of gearing to another

 Calculating a project beta factor

 Estimating an equity beta factor for a private company

7 Asset beta

7.1 Changes in gearing, the cost of equity and equity beta factors

Modigliani and Miller’s theories are used to demonstrate that as the level of gearing

in a company increases, assuming no change in business risk, the cost of equity also

increases. This also means that as gearing increases, the beta factor of the company’s

equity must also increase, since there is no change in the risk-free rate of return or

the returns on the market portfolio.

Companies that are otherwise identical except for their level of gearing will

therefore have different costs of equity and different equity beta factors.

Similarly, if a company changes its gearing level without changing its business risk

profile, there will be a change in its cost of equity and equity beta factor.

7.2 Definition of asset beta

An asset beta is the beta factor that would be applied to a company’s assets if

financing risk is removed from the calculation. It is therefore a beta factor that

applies to entire companies or individual projects. It provides a measure of the

underlying business risk faced by the entire firm regardless of its financing

structure.

An asset beta is sometimes called unlevered beta or ungeared beta, and could be

described as the beta of equity in a company that is totally equity-financed.

7.3 The asset beta formula

Modigliani and Miller argued that there is a difference between the beta factor in an

ungeared (all equity) company and the beta factor in a similar geared company. The

beta factor in a totally ungeared company (the ungeared beta) is the asset beta for

the firm.

Paper P4: Advanced financial management

The formula, adjusted for taxation, is:

()

(

)

()

t1DE

t1D

t1DE

DEGA

−+

−

×+

−+

×=

βββ

where

= the asset beta or ungeared beta

= the beta of the equity in a geared company

= the beta of the debt capital in the geared company

E = the market value of the equity in the geared company

D = the market value of the debt capital in the geared company

t = the rate of taxation on profits (the rate of tax relief on interest payments on debt

capital).

This formula is included in the formula sheet for your examination. You do not

need to learn it, but you must be able to apply it.

For the purpose of examination questions, it is often assumed that debt capital in a

company is risk-free. This is a simplifying assumption which makes the calculations

much easier!

If it is assumed that debt is risk-free, the asset beta formula simplifies to:

()

t1DE

EGA

−+

×=

ββ

Modigliani and Miller therefore argued that the beta factor in an all-equity company

is lower than the beta factor in an identical but geared company, by a factor of

E + D1− t

()

[]

Example

A company is all-equity financed, and its equity beta factor is 1.05. Its total market

value is $240 million. This includes debt capital with a market value of $80 million.

It is assumed that the debt capital is risk-free. The rate of tax is 30%.

Required

Calculate the asset beta for the company.

ACCA P4 Advanced Financial Management - 2010 - Study text - Emile Woolf Publishing

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