ACCA P4 Advanced Financial Management - 2010 - Study text - Emile Woolf Publishing
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Chapter 6: Cost of capital
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Calculating the WACC
Method of calculating the WACC
WACC and market value
4 Calculating the WACC
4.1 Method of calculating the WACC
The WACC is a weighted average of the (after-tax) cost of all the sources of capital
for the company. It is calculated as follows:
Source of finance Market value
×
Cost Market value × Cost
r MV × r
Equity MV
E
×
K
E
MV
E
× K
E
Preference shares MV ×
K
P
MV
P
× K
P
Debt MV
D
×
K
D
MV
D
× K
D
Total
Σ
MV
ΣMV x r
WACC =
Σ
MV x r
Σ
MV
The WACC for a company is found using the method shown above. K
D
is the after-
tax cost of debt capital. If there is more than one source of debt, each with a different
cost, you can either:
have a separate line in the table for each item of debt, or
calculate a weighted average cost of all debt capital first (after-tax), then have
just one line in the table with the total market value of all debt capital and the
weighted average after-tax cost of the debt.
Example
A company has 8 million shares each with a value of $7.90, whose cost is 8.4%. It has
6% bonds with a market value of $50 million and an after-tax cost of 3.6%. It has a
bank loan of $10 million whose after-tax cost is 4.1%. It also has 2 million 8%
preference shares of $1 whose market price is $1.33 per share and whose cost is 6%.
Calculate the WACC.
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Answer
Source of finance Market value Cost Market value × Cost
$ million r % MV × r
Equity 63.20
8.4
530.88
Preference shares 2.66
6.0
15.96
Bonds 50.00
3.6
180.00
Bank loan 10.00
4.1
41.00
125.86
767.84
WACC =
6.10%
125.86
767.84
=
Example
You should be prepared in your examination to calculate the cost of equity in a
company, given the WACC, the cost of debt and the market value of equity and
debt. Here is an example.
A company has a weighted average cost of capital of 8.2%. It has 100 million shares
currently valued at $6 each, and it has a bank loan of $200 million on which the rate
of interest is 6.5%. The rate of taxation is 30%.
Required
Calculate the current cost of equity in the company.
Answer
The cost of equity can be calculated as the solution to a mathematical problem.
Let the cost of equity be r.
The after-tax cost of debt is 6.5% × (1 – 0.30) = 4.55%.
The WACC , which is 8.2%, is calculated as follows:
Source of finance Market
value
Cost
Market value
× Cost
$ million
%
MV × r
Equity 600.0
r
600r
Debt 200.0
4.55
910.0
800.0
600r + 910
WACC = 8.2% =
%
800
910 600r
+
8.2 (800) = 600r + 910
600r = 5,650
r = 9.417%
The cost of equity is 9.417%, say 9.4%.
Chapter 6: Cost of capital
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4.2 WACC and market value
For a company with constant annual ‘cash profits’, there is an important connection
between WACC and market value. (Note: ‘Cash profits’ are cash flows generated
from operations, before deducting interest costs.)
If we assume that annual cash profits are a constant amount in perpetuity, the total
value of a company, equity plus debt capital, is calculated as follows:
Total market value of the company =
Annual cash profits
WACC
From this formula, the following conclusions can be made:
The lower the WACC, the higher the total value of the company will be (equity +
debt capital), for any given amount of annual profits.
Similarly, the higher the WACC, the lower the total value of the company.
For example, if annual cash profits are $12 million, the total market value of the
company would be:
$100 million if the WACC is 12% ($12 million/0.12)
$120 million if the WACC is 10% ($12 million/0.10)
$200 million if the WACC is 6% ($12 million/0.06).
The
aim should therefore be to achieve a level of financial gearing that minimises
the WACC
, in order to maximise the value of the company.
Important questions in financial management are:
How does the WACC change with changes in gearing?
For each company, is there an ‘ideal’ level of gearing that minimises the WACC?
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Cost of capital and gearing
The traditional view of gearing and WACC
The Modigliani-Miller propositions: ignoring corporate taxation
The Modigliani-Miller propositions: allowing for corporate taxation
5 Cost of capital and gearing
For a given level of annual cash profits before interest and tax, the value of a
company (equity + debt) is maximised at the level of gearing where WACC is
lowest. This should also be the level of gearing that optimises the wealth of equity
shareholders.
The question is therefore:
How does a change in gearing affect the WACC, and is there a level of gearing
where the WACC is minimised?
The most important analysis of gearing and the cost of capital, for the purpose of
your examination, is the analysis provided by Modigliani and Miller that allows for
tax relief on debt interest.
However, the traditional view of WACC and gearing, and Modigliani and Miller’s
propositions ignoring tax relief on debt are also described briefly.
5.1 The traditional view of gearing and WACC
The traditional view of gearing is that there is an optimum level of gearing for a
company. This is the level of gearing at which the WACC is minimised.
As gearing increases, the cost of equity rises. However, as gearing increases,
there is a greater proportion of debt capital in the capital structure, and the cost
of debt is cheaper than the cost of equity. Up to a certain level of gearing, the
effect of having more debt capital has a bigger effect on the WACC than the
rising cost of equity, so that the WACC falls as gearing increases.
However, when gearing rises still further, the increase in the cost of equity has a
greater effect than the larger proportion of cheap debt capital, and the WACC
starts to rise.
The traditional view of gearing is therefore that an optimum level of gearing exists,
where WACC is minimised and the value of the company is maximised.
Chapter 6: Cost of capital
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Traditional view of gearing and the WACC
5.2 The Modigliani-Miller propositions: ignoring corporate taxation
The traditional view of gearing and WACC was challenged by Modigliani and
Miller in the 1950s. Initially, their arguments were based on the assumption that
corporate taxation, and the tax relief on interest, could be ignored.
You do not need to know Modigliani and Miller’s arguments, only the conclusions
they reached. They argued that if corporate taxation is ignored, an increase in
gearing will have the following effect:
As the level of gearing increases, there is a greater proportion of cheaper debt
capital in the capital structure of the firm.
However, the cost of equity rises as gearing increases.
As gearing increases, the net effect of the greater proportion of cheaper debt and
the higher cost of equity is that the WACC remains unchanged.
The WACC is the same at all levels of financial gearing.
The total value of the company is therefore the same at all levels of financial
gearing
Modigliani and Miller therefore reached the conclusion that the level of gearing is
irrelevant for the value of a company. There is no optimum level of gearing that a
company should be trying to achieve.
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Modigliani-Miller view of gearing and the WACC: no taxation
Modigliani and Miller’s propositions: ignoring taxation
Modigliani and Miller’s arguments, ignoring taxation, can be summarised as two
propositions.
Proposition 1. The WACC is constant at all levels of gearing. For companies
with identical annual profits and identical business risk characteristics, their
total market value (equity plus debt) will be the same regardless of differences in
gearing between the companies.
Proposition 2. The cost of equity rises as the gearing increases. The cost of equity
will rise to a level such that, given no change in the cost of debt, the WACC
remains unchanged.
Modigliani-Miller formulae: ignoring taxation
There are three formulae for the Modigliani and Miller theory, ignoring corporate
taxation. These are shown below. The letter ‘
U
’ refers to an ungeared company (all-
equity company) and the letter ‘
G
’ refers to a geared company.
(1)
WACC
The WACC in a geared company and the WACC in an identical but ungeared
(all-equity) company are the same:
WACC
G
= WACC
U
This formula expresses a part of proposition 1.
(2)
Total value of the company (equity plus debt capital)
The total value of an ungeared company is equal to the total value of an
identical geared company (combined value of equity + debt capital):
V
G
= V
U
This formula expresses another part of proposition 1.
Chapter 6: Cost of capital
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(3) Cost of equity
The cost of equity in a geared company is higher than the cost of equity in an
ungeared company, by an amount equal to:
the difference between the cost of equity in the ungeared company and the
cost of debt (K
EU
– K
D
)
multiplied by the ratio of the market value of debt to the market value of
equity in the geared company (D/E).
K
EG
= K
EU
+
D
E
K
EU
−K
D
()
This formula expresses proposition 2.
Example
An all-equity company has a market value of $150 million and a cost of equity of
10%. It borrows $50 million of debt finance, costing 6%, and uses this to buy back
and cancel $50 million of equity. Tax relief on debt interest is ignored.
Required
According to Modigliani and Miller, if taxation is ignored, what would be the effect
of the higher gearing on (a) the WACC (b) the total market value of the company
and (c) the cost of equity in the company?
Answer
According to Modigliani and Miller:
(a)
WACC. The WACC in the company is unchanged, at 10%.
(b)
Total value. The total market value of the company with gearing is identical
to the market value of the company when it was all equity, at $150 million.
This now consists of $50 million in debt and $100 million equity ($150 million
– $50 million of debt)
(c)
Cost of equity. The cost of equity in the geared company is
()
12.0%%610
100
50
10% =
⎥
⎦
⎤
⎢
⎣
⎡
−×+
Example
A company has $500 million of equity capital and $100 million of debt capital, all at
current market value. The cost of equity is 14% and the cost of the debt capital is 8%.
The company is planning to raise $100 million by issuing new shares. It will use the
money to redeem all the debt capital.
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Required
According to Modigliani and Miller, if the company issues new equity and redeems
all its debt capital, what will be the cost of equity of the company after the debt has
been redeemed? Assume that there is no corporate taxation.
Answer
In the previous example, the Modigliani-Miller formulae were used to calculate a
cost of equity in a geared company, given the cost of equity in the company when it
is ungeared (all-equity). This example works the other way, from the cost of equity
in a geared company to a cost of equity in an ungeared company. The same
formulae can be used.
Using the known values for the geared company, we can calculate the cost of equity
in the ungeared company after the debt has been redeemed.
K
EG
= K
EU
+ D/E [K
EU
-
K
D
]
14.0 = K
EU
+ 100/500 [K
EU
– 8.0]
1.2 K
EU
= 14.0 + 1.6
K
EU
= 13.0% (15.6/1.2).
5.3 The Modigliani-Miller view: allowing for corporate taxation
Modigliani and Miller revised their arguments to allow for the fact that there is tax
relief on interest. You do not need to know the arguments they used to reach their
conclusions, but you must know what their conclusions were. You should also
know and be able to apply the formulae described below.
(The formula for the cost of equity is given in the formula sheet in your
examination, so you do not need to learn it.)
Modigliani and Miller argued that allowing for corporate taxation and tax relief on
interest, an increase in gearing will have the following effect:
As the level of gearing increases, there is a greater proportion of cheaper debt
capital in the capital structure of the firm. However, the cost of equity rises as
gearing increases.
As gearing increases, the net effect of the greater proportion of cheaper debt and
the higher cost of equity is that the WACC becomes lower. Increases in gearing
result in a reduction in the WACC.
The WACC is therefore at its lowest at the highest practicable level of gearing.
(There are practical limitations on gearing that stop it from reaching very high
levels. For example, lenders will not provide more debt capital except at a much
higher cost, due to the high credit risk).
Chapter 6: Cost of capital
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The total value of the company is therefore higher for a geared company than for
an identical all-equity company. The value of a company will rise, for a given
level of annual cash profits before interest, as its gearing increases.
Modigliani and Miller therefore reached the conclusion that because of tax relief on
interest, there is an optimum level of gearing that a company should be trying to
achieve. A company should be trying to make its gearing as high as possible, to the
maximum practicable level, in order to maximise its value.
Modigliani-Miller view of gearing and the WACC: with taxation
Modigliani and Miller’s propositions: allowing for taxation
Modigliani and Miller’s arguments, allowing taxation, can be summarised as two
propositions.
Proposition 1. The WACC falls continually as the level of gearing increases. In
theory, the lowest cost of capital is where gearing is 100% and the company is
financed entirely by debt. (Modigliani and Miller recognised, however, that
‘financial distress’ factors have an effect at high levels of gearing, increasing the
cost of debt and the WACC.) For companies with identical annual profits and
identical business risk characteristics, their total market value (equity plus debt)
will be higher for a company with higher gearing.
Proposition 2. The cost of equity rises as the gearing increases. There is a
positive correlation between the cost of equity and gearing (as measured by the
debt/equity ratio).
Modigliani-Miller formulae: allowing for taxation
There are three formulae for the Modigliani and Miller theory, allowing for
corporate taxation. These are shown below. The letter ‘
U
’ refers to an ungeared
company (all-equity company) and the letter ‘
G
’ refers to a geared company.
(1)
WACC
The WACC in a geared company is lower than the WACC in an all-equity
company, by a factor of
1−
Dt
D +E
()
.
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WACC
G
= WACC
U
1−
Dt
D +E
()
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
This formula expresses a part of proposition 1.
(2)
Value of a company
The total value of a geared company (equity + debt) is equal to the total value
of an identical ungeared company plus the value of the ‘tax shield’. This is the
market value of the debt in the geared company multiplied by the rate of
taxation (Dt).
V
G
= V
U
+Dt
where:
V
G
= value of geared company
V
U
= value of an identical but ungeared (all-equity) company
D = market value of the debt in the geared company
t = the rate of taxation on company profits.
This formula expresses another part of proposition 1.
(3)
Cost of equity
The cost of equity in a geared company is higher than the cost of equity in an
ungeared company, by a factor equal to:
the difference between the cost of equity in the ungeared company and the
cost of debt, (K
EU
– K
D
)
multiplied by the ratio
1−t
()
×
D
E
.
()
E
D
KK)1(KK
DEUEUEG
−−+= t
This formula expresses proposition 2. It is given to you in your examination,
in a formula sheet. Although you do not need to learn the formula, you should
become familiar with it, and know how to use it.
Example
An all-equity company has a market value of $60 million and a cost of equity of 8%.
It borrows $20 million of debt finance, costing 5%, and uses this to buy back and
cancel $20 million of equity. The rate of taxation on company profits is 25%.
According to Modigliani and Miller:
(a)
Market value
The market value of the company after the increase in its gearing will be:
V
G
= V
U
+Dt