Transcript Recovering Stochastic Processes from Option Prices

1
Capital Structure
Capital Structure
Francesca Cornelli
London Business School
[email protected]
2
Capital Structure
OUTLINE



Does Capital structure Matter?
No: The Modigliani and Miller propositions.
Yes:





Corporate Taxes
Personal Taxes
Costs of Financial Distress
Agency Costs.
How to take into account corporate taxes when valuing a
project:


APV
WACC.
3
Capital Structure
Definition

The Capital Structure of a firm is the mix of different securities
issued by the firm to finance its projects. Examples of such
securities are





bonds
bank debt
common stocks
etc…
Does Capital structure Matter?

We will focus on the consequences of the choice between different
proportions of debt and equity in order to finance a given level of
assets.
4
Capital Structure
Firm’s Objective
Given the firm’s assets and investment plan, find the debt
proportion that maximizes firm value
Market Value - balance sheet
Assets
Debt (D)
Equity (E)
Firm value (V)
5
Capital Structure

A change in the capital structure of the firm that leaves the
assets of the firm unchanged will not change X, the cash flows
generated by the assets of the firm.

So, one might be tempted to think that the capital structure of
a firm does not alter its value.
Yet, recall that

 D  E
I
D 
rD
V
E 

X  I
rE
As rD and rE are not equal, the changes in D and E brought
about by a change in I will not cancel each other.
6
Capital Structure
Example 1
Consider a firm with projects that expect to generate £10,000
in perpetuity.
X
I
rD
rE




10,000
2,000
5%
10%
Now, let I= £5,000
What is Wrong?
2,000
D
 40,000
0.05
8,000
E
 80,000
0.1
V  D  E  120,000
5,000
D
 100,000
0.05
5,000
E
 50,000
0.1
V  D  E  150,000
7
Capital Structure
M&M Proposition I

Modigliani & Miller




Under the following assumptions:
No taxes
No costs of financial distress
Individuals can borrow and lend on the same terms as corporations
No asymmetry of information

The value of a firm is independent of its capital structure

These assumptions are clearly not reasonable. Does that
make MMI useless?
8
Capital Structure
MM Proposition I

NO

What MMI indicates is that if capital structure matters, as it
does indeed, it must be because of taxes, the costs of
financial distress or differences in the lending and borrowing
terms offered to corporations and individuals.

Other reasons exists for capital structure to matter. These will
be discussed later.
9
Capital Structure
Example
A firm has assets of £1 million and is 50% debt-financed. Cost
of debt is 6%.
State
Prob. Assets PBIT Debt
Good
Bad
0.5
0.5
1.20m
1.05m
.20m .53m
.05m .53m
Interest PBT Equity
0.03m
0.03m
.17m .67m
.02m .52m
The manager thinks he can increase the value of the firm by
retiring debt. He thinks some investors might value the
resulting decrease in riskiness of the equity.
State Prob.
Assets = Equity
PBIT = PBT
Good 0.5
1.2m
.2m
Bad
0.5
1.05m
.05m
10
Capital Structure
Unlevered Firm
60
50
Cash Flow
40
30
CF to Equity
20
10
0
1
2
3
4
5
6
7
State of the worls
8
9
10
11
12
13
11
Capital Structure
Levered Firm with "Safe Debt"
60
50
Cash Flow
40
30
CF to Equity
20
10
CF to Debt
0
1
2
3
4
5
6
7
State of the World
8
9
10
11
12
13
12

Capital Structure
Let VLdenote the value of the geared (or levered) firm and VU
that of the ungeared (or unlevered) firm. We shall show that
V L VU
13
Capital Structure
Example (continue)

Consider buying 1% of the unlevered firm. The payoff is:




0.01 of £1.2m= £12,000 in the good state
0.01 of £1.05m= £10,500 in the bad state
The cost is 0.01 X VU
Consider buying 1% of the levered firm (1% of its debt and 1%
of its equity). The payoff is



0.01 of £0.67m+0.01 of 0.53m = £12,000 in the good state
0.01 of £0.52m+0.01 0f 0.53m= £10,500 in the bad state
0.01E  0.01D  0.01VL
The cost is
Since the payoff in each state is the same, we conclude that
the cost must be the same

Why?
14





Capital Structure
No investor would buy a more expensive investment that has
the same payoff as a cheaper investment
Thus, the value of the geared firm must equal that of the allequity firm.
By buying 1% of the geared firm’s equity and 1% of its debt,
the investor who chose to do so was able to obtain the same
payoffs as those of the investor who chose to buy 1% of the
unlevered firm’s equity.
In other words, the investor in the geared firm was able to
offset the greater riskiness of the equity of the geared firm by
also holding its debt: she unlevered her shares of the geared
firm on her own
Because the investor could unlever her shares on her own,
there was no need for the firm itself to do so, and therefore no
gain for the firm for doing it.
15
Capital Structure
How leverage Affects Risk and Returns



Leverage increases the variability and the expected return per
share.
M&M I tells us that the price of the share does not change.
This is possible if the increase in the expected return is exactly
offset by the increase in risk.
Since the expected return on a portfolio is equal to a weighted
average of the expected returns on the individual holdings we
have:
D
rE  rA  (rA  rD )
E
This is M&M’s proposition II.
16
Capital Structure
17
Capital Structure
Example (continued)
A 1

The asset beta of the unlevered firm is

As the risk free rate is 6% and the risk premium is 6.5%, the
required return on equity is therefore 12.5%

The value of the unlevered firm is
0.5 1,200 ,000  0.5 1,050 ,000
VU 
 1m
1.125
18
Capital Structure

The beta of the equity of the geared firm can be shown to be
equal to 2, making the required return on equity equal to 19%

The value of the debt is
530,000
D
 500 ,000
1.06
The value of the equity is

0.5  670 ,000  0.5  520 ,000
E
 500 ,000
1.19

We therefore have
VL  D  E  1,000,000  VU
19
Capital Structure
20
Capital Structure
21
Capital Structure
How leverage Affects Risk and Returns

Also the beta of a firm is a weighted average of the betas of
the individual securities.
 D
  E

A  
 D   
 E 
DE
 DE

D
  E   A  ( A   D )
E
22
Capital Structure
Leverage and Earning per Share




EPS is profit (or net earnings) divided by number of
outstanding shares.
An increase in EPS can be the consequence of an
improvement in firm performance (good news).
It can also be achieved, however, by leverage. The expected
EPS increases with leverage. It represents only the increased
compensation required by shareholders for the additional risk
they have bear (no news).
What happens to the price-earning ratio (P/E)?
23
Capital Structure
Example (continued)

Suppose the geared firm had 100,000 shares, each selling for
£5 (recall that E = £500,000).

Suppose the good state occurs, and profit is £200,000 £30,000 = £170,000.

We therefore have
170,000
EPS 
 1.7
100,000
24
Capital Structure

Now suppose the all-equity firm had retired debt by issuing
100,000 shares at £5 each.

The all-equity firm therefore has a total of 200,000 shares.

In the good state, its PBT is £200,000 and we have
200,000
EPS 
1
200,000

The EPS of the all-equity firm is lower. Is the unlevered firm
less valuable?
25



Capital Structure
No
We know that VL  VU . The decrease in EPS is simply a
consequence of the decrease in the risk borne by equity.
Furthermore, note that the decrease in gearing not only
decreases EPS (actual in the good state and expected), but
also increases the price-earnings ratio (the price of a share
remains the same, and EPS decreases).
26
Capital Structure
27
Capital Structure
28
Capital Structure Does matter:
Corporate taxes

Capital Structure
The interest that a company pays is tax deductible, while
dividends and retained earnings are not.
$1 Operating Income
Corporate tax
None
Tc
Income after
tax
$1
$(1 - Tc )
To
Bondholders
To
Stock Holders
29
Capital Structure
Example

Consider an all-equity firm that has assets of £ 1 million and
expected EBIT of £ 200,000 per year. It expects to pay tax of
£70,000 (the corporate tax rate is 35%) so it has net income to
shareholders of £130,000.

Let the firm issue debt to finance a £ 500,000 repurchase of
equity. The debt pays interest of £ 30,000 (6% interest rate).
30
Capital Structure
Example (continued)
EBIT
Interest
Unlevered
200,000
0
30,000
Income
200,000
170,000
Tax
70,000
59,500
Net Combined Income 130,000

Levered
200,000
110,500+30,000=140,500
Debt provides Tax Shield to the shareholders of 30,000 from
tax: a saving of 0.35 X 30,000 = 10,500
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Capital Structure
Tax Shield

The value of a levered firm is no longer equal to that of an
unlevered firm, but is greater by an amount that represents the
present value of the tax shield provided by debt
VL  VU  PV(tax shield)

Every year the tax shield is Tc rd D .

To calculate the present value of the tax shield, which discount
rate should we use?.
32
Capital Structure
Example (continued):

Assume that the debt is permanent. The yearly tax shield is
£10,500, and thus the present value of the tax shield is
10,500
 175,000
0.06

In general, when debt is permanent, we have
Tc rd D
PV(Tax Shield) 
 Tc D
rd

We have VL  VU  PV(Tax Shield) . So leverage increases
value.
Why not go for 100% debt?
33
Capital Structure
Unlevered Firm with Corporate Taxes
60
50
Cash Flow
40
30
CF to Equity
20
10
CF to Taxes
0
1
2
3
4
5
6
7
State of the World
8
9
10
11
12
13
34
Capital Structure
Levered Firm with "Safe Tax" and Corporate Taxes
60
50
Cash Flow
40
CF to Equity
30
20
CF to Taxes
10
CF to Debt
0
1
2
3
4
5
6
7
State of the World
8
9
10
11
12
13
Microsoft Balance Sheets
In millions
June 30
1998
$ 8,966
980
427
$13,927
1,460
502
10,373
1,465
2,346
203
15,889
1,505
4,703
260
$14,387
$22,357
Total current assets
Property and equipment
Equity investments
Other assets
Total assets
Liabilities and stockholders’ equity
Current liabilities:
Accounts payable
Accrued compensation
Income taxes payable
Unearned revenue
Other
Total current liabilities
Commitments and contingencies
Stockholders’ equity:
Convertible preferred stock – shares authorized 100;
shares issued and outstanding 13
Common stock and paid-in capital – shares authorized 8,000;
shares issued and outstanding 2,408 and 2,470
Retained earnings
Total stockholders’ equity
Total liabilities and stockholders’ equity
$
721
336
466
1,418
669
$
759
359
915
2,888
809
3,610
5,730
980
980
4,509
5,288
8,025
7,622
10,777
16,627
$14,387
$22,357
Source http://www.microsoft.com/msft/ar98/downloads/msftar98.doc
Assets
Current assets:
Cash and short-term investments
Accounts receivable
Other
1997
36
Capital Structure
Capital budgeting

Before looking at the effect of personal taxes and costs of
financial distress, we want to see how to take into account
corporate taxes when valuing a firm or a project

In fact, if in presence of corporate taxes leverage affects
value, then we want to see how to take it into account.
37
Capital Structure
Adjusted Present Value

The relation
VL  VU  TC D
which relates the value of a levered firm to that of an
unlevered firm that is otherwise identical to the former
suggests that the value of a levered firm can be obtained by



determining the value of the levered firm as if it were unlevered, and
adjusting the obtained value for the presence of the tax shield
Such an approach is called Adjusted Present Value. It is valid
for NPV as well as PV.
38
Capital Structure

The concept of Adjusted Present Value is very general.

It can be used to account for the value of the tax shields
provided by various types of debt, for issuing costs, for the
costs of financial distress, etc...
39
Capital Structure
Weighted Average Cost of Capital

As an alternative to adjusting the PV of a levered firm for the
presence of a tax shield, it is possible to adjust the discount
rate that is used to discount the cash flows of the firm.

More specifically, the WACC, which was previously defined as
D
E
WACC 
rD 
rE
ED
ED
becomes, in the presence of corporate taxes
D
E
WACC 
(1  Tc )rD 
rE
ED
ED
(1  TC )rD is used in place of r
D

Why the lower interest rate in the case of taxes?
40

Capital Structure
The factor (1  TC ) by which the interest rate rD is multiplied
reflects the fact that interest payments provide a tax shield.

The WACC should be used only in cases where the ratio of
debt to the market value of the firm is constant (in addition to
the conditions specified in previous lectures).

It is often used as an approximation in the case where the
ratio of debt to the market value of the firm is not constant (but
the other conditions are nonetheless true).

Under the above conditions, one can either


explicitly account for the value of the tax shield, by using APV, or
use the WACC, in which case no tax shield is to be added.
41
Capital Structure
How leverage affects the betas

The relation between assets, equity and debt betas
D
E
A 
D 
E
DE
DE
remains true in the presence of corporate taxes in the case
where the risk of the tax shield is equal to the risk of the
assets.

It becomes
(1  TC ) D
E
A 
D 
E
E  (1  TC ) D
E  (1  TC ) D
in the case where the risk of the tax shield is equal to the risk
of the debt.
42
Capital Structure
EXAMPLE


A firm considers a project that requires an initial investment of
£10m and has cash flow of £2m in perpetuity.
The firm has cost of equity rE =15%, cost of debt rD  rf =5%,
and is 50% debt-financed (debt equals 50% of the value of the
firm, equity equals 50%).

The project is in all respects similar to the present operations
of the firm, and will also be 50% debt-financed.

NB: When we talk about the value of a company we usually
mean its present value. However in this case the firm is
considering whether to undertake a project and therefore it is
computing the net present value of the project.
43

Capital Structure
The WACC of the firm is
D
D
WACC 
(1  TC )rD 
rE
DE
DE
 0.5  0.66 0.05  0.5  0.15 
 0.0915or 9.15%

The project therefore has NPV
2  0.66
NPV  10 
 4.43m
0.0915


Note that yearly cash flow was not adjusted to account for the
presence of the tax shield.
The WACC does so.
44


Capital Structure
What if we use the APV?
Let rM =10%.  E can therefore be obtained from the CAPM
rE  rf   E (rM  rf )
 0.15  0.05   E (0.10  0.05)
 E  2

 A can now be obtained from  E by using
A 

(1  TC ) D
E
D 
E
E  (1  TC ) D
E  (1  TC ) D
2
 A 
 1.2
1  (1  0.34)
We therefore have
rA  0.05  1.2(0.10  0.05)  0.11  11%
45

Capital Structure
The NPV of the project assumed to be all-equity financed is
2  0.66
NPV(all equity)  10 
 2m
0.11

The NPV of the project, with 50% debt, (ie the APV) is given
by the NPV(all equity) + PV(Tax Shield)

The present value of the tax shield provided by debt is
TC D

But what is D?
46
Capital Structure

We know that debt equals 50% of the present value of the
project.

The present value of the project equals the sum of the value of
the fixed assets of the project, the net present value of the
project and the present value of the tax shield.

In other words, we have
D

PV(project with 50% debt) 10  2  0.34 D

2
2
The APV therefore equals
APV  2  0.34  7.23  2  2.46  4.46 m
which is the same as the NPV obtained with the WACC (save
for some rounding errors).
47
Capital Structure
Back to the optimal capital structure

Now that we have seen how to take into account corporate
taxes, let us go back to see personal taxes and costs of
financial distress.

We have already seen that these can be easily taken into
account in the APV
48
Capital Structure
Corporate and Personal Taxes
$1 Operating Income
Corporate tax
None
TC
Income after
corporate tax
$1
$1  TC
Personal tax
TP
TPE (1  TC )
Income after
taxes
$1  TP
(1  TPE )(1  TC )
To bondholders
To shareholders
49
Capital Structure
Corporate and Personal Taxes

The tax shield (per $1 paid interest rate) is
Corporate tax+personal tax on equity - personal tax (on interest)

In the case of perpetual debt, the present value of the tax
shield is (we use rD (1  TP ) as a discount factor):
 (1  TC )(1  TPE ) 
1 
D
(1  TP )


and
 (1  TC )(1  TPE ) 
VL  VU  1 
D
(1  TP )


where TC is the corporate tax rate, TP is the marginal rate of
personal tax, and TE is effective tax rate on equity.
50
Capital Structure
Example

Consider a company who pays no dividends, and its
shareholders are top rate taxpayer (personal tax 40%), who do
not intend to realize capital gain in the near future. Given the
option to defer taxes on capital gains, they view the effective
tax rate on capital gain as 5%. The present value of the tax
shield is
0.675 
 (1  0.35)(1  0.05) 

1 
 D  1 
D  0
(1  0.4)
0.6 



Leverage now decreases value.
51
Capital Structure
Capital Structure Does matter:
The Cost of Financial Distress
Levered Firm with "Risky Debt" and Corporate Taxes
60
CF to Equity
50
CF to Taxes
Cash Flow
40
30
CF to Debt
20
10
0
1
2
3
4
5
6
7
State of the World
8
9
10
11
12
13
52
Capital Structure Does matter:
The Cost of Financial Distress
Capital Structure

A firm that is unable to meet its debt obligations or that will be
unable to do so at some point in the near future is said to be in
Financial Distress.

A firm in financial distress can seek protection from its
creditors by filing for Bankruptcy. It is then either liquidated or
reorganized.
53
Capital Structure Does matter:
The Cost of Financial Distress


In some industries, liquidation value is low (for example
software houses), and for some it is high (For example the
liquidation value of an airline is very high).
There are also indirect costs of reorganizing a bankrupt firm:




Capital Structure
puts severe constraints on the ability of management to conduct
business.
Hampers the relations of the firm with its customers and its suppliers.
Leads to loss of human capital and of growth opportunities.
In some cases, indirect costs are estimated to be as high as
20% of the value of the firm.
54
Capital Structure
Financial Distress Without Bankruptcy

There are yet more costs to financial distress than those of
liquidation or reorganisation.

These costs are borne by firms which are neither bankrupt nor
insolvent, as well as by bankrupt firms that are being
reorganised.

They arise from the severe conflicts of interest between
shareholders and bondholders created by financial distress.
55
Capital Structure
Conflict of interest
Example

Consider a firm with £ 50 of 1-year debt.
BOOK
NWC 10
MARKET
50
D
50
E
FA
90
TA
100 100 T L
NWC 10 30 D
FA
30 10 E
V
40 40 V
56
Capital Structure
Incentives for Costly Games
A) The shareholders of the firm favor risky projects.
t 0
t 1
40
Prob.
1/5
0
4/5
- 10
B) Shareholders favor high dividends. They gain the full
benefit, but decline in the firm value is shared with
bondholders.
57
Capital Structure
Capital Structure Does matter:
The Cost of Financial Distress
Levered Firm with "Risky Debt", Corporate Taxes, and
Financial Distress Costs
Cash Flow
60
50
CF to Equity
40
CF to Taxes
30
CF to Debt
20
10
Financial Distress Cost
0
1
2
3
4
5
6
7
State of the World
8
9
10
11
12
13
58
Capital Structure Does matter:
The Cost of Financial Distress


Capital Structure
Increasing leverage increases the probability of default, and
with it the PV of the expected costs of financial distress.
Thus, leverage reduces the market value of the firm by
increasing the present value of financial distress.
VL  VU  PV(tax shield) - PV(costs of financial distress)
Who has to absorb the costs of financial distress?
Of course, when the firm is in financial distress, it is the debt
holders that incur the costs.
However, at the time of issuing the debt, it is the shareholders
that incur the present value of the costs of financial distress.
59
Capital Structure
Summary
Market Value of The Firm
Maximum value of firm
Costs of
financial distress
PV of interest
tax shields
Value of
unlevered
firm
Optimal amount
of debt
Debt
Value of levered firm