Recovering Stochastic Processes from Option Prices

1
Capital Structure
Francesca Cornelli London Business
School fcornelli_at_lbs.ac.uk
2
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
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
Firms Objective

• Given the firms 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

• 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
• As and are not equal, the changes in D
  and E brought about by a change in I will not
  cancel each other.

6
Example 1

• Consider a firm with projects that expect to
  generate 10,000 in perpetuity.

Now, let I 5,000
What is Wrong?
7
MM 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
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
Example

• A firm has assets of 1 million and is 50
  debt-financed. Cost of debt is 6.
• State Prob. Assets PBIT Debt Interest PBT
  Equity
• Good 0.5 1.20m .20m .53m 0.03m .17m
  .67m
• Bad 0.5 1.05m .05m .53m 0.03m .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
CF to Equity
11
CF to Equity
CF to Debt
12

• Let denote the value of the geared (or
  levered) firm and that of the ungeared (or
  unlevered) firm. We shall show that

13
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
• Consider buying 1 of the levered firm (1 of its
  debt and 1 of its equity). The payoff is
• 0.01 of 0.67m0.01 of 0.53m 12,000 in the
  good state
• 0.01 of 0.52m0.01 0f 0.53m 10,500 in the bad
  state
• The cost is
• Since the payoff in each state is the same, we
  conclude that the cost must be the same
• Why?

14

• 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 all-equity firm.
• By buying 1 of the geared firms 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
  firms 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
How leverage Affects Risk and Returns

• Leverage increases the variability and the
  expected return per share.
• MM 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
• This is MMs proposition II.

16
(No Transcript)
17
Example (continued)

• 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

18

• 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
• The value of the equity is
• We therefore have

19
(No Transcript)
20
(No Transcript)
21
How leverage Affects Risk and Returns

• Also the beta of a firm is a weighted average of
  the betas of the individual securities.

22
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
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

24

• 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
• The EPS of the all-equity firm is lower. Is the
  unlevered firm less valuable?

25

• No
• We know that . 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
(No Transcript)
27
(No Transcript)
28
Capital Structure Does matterCorporate taxes

• The interest that a company pays is tax
  deductible, while dividends and retained earnings
  are not.

Corporate tax
Income after tax
29
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
Example (continued)

• Unlevered Levered
• EBIT 200,000 200,000
• Interest 0 30,000
• Income 200,000 170,000
• Tax 70,000 59,500
• Net Combined Income 130,000 110,50030,000140
  ,500
• Debt provides Tax Shield to the shareholders of
  30,000 from tax a saving of 0.35 X 30,000
  10,500

31
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
• Every year the tax shield is .
• To calculate the present value of the tax shield,
  which discount rate should we use?.

32
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
• In general, when debt is permanent, we have
• We have
  . So leverage increases value.
• Why not go for 100 debt?

33
CF to Equity
CF to Taxes
34
CF to Equity
CF to Taxes
CF to Debt
35
Microsoft Balance Sheets
Source http//www.microsoft.com/msft/ar98/download
s/msftar98.doc
36
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
Adjusted Present Value

• The relation
• 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

• 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
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
• becomes, in the presence of corporate taxes
• is used in place of
• Why the lower interest rate in the case of taxes?

40

• The factor by which the interest
  rate 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
How leverage affects the betas

• The relation between assets, equity and debt
  betas
• 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
• in the case where the risk of the tax shield is
  equal to the risk of the debt.

42
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 15, cost of
  debt 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

• The WACC of the firm is
• The project therefore has NPV
• Note that yearly cash flow was not adjusted to
  account for the presence of the tax shield.
• The WACC does so.

44

• What if we use the APV?
• Let 10. can therefore be obtained
  from the CAPM
• can now be obtained from by using
• We therefore have

45

• The NPV of the project assumed to be all-equity
  financed is
• 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
• But what is D?

46

• 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
• The APV therefore equals
• which is the same as the NPV obtained with the
  WACC (save for some rounding errors).

47
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
Corporate and Personal Taxes
Corporate tax
Income after corporate tax
Personal tax
Income after taxes
49
Corporate and Personal Taxes

• The tax shield (per 1 paid interest rate) is
• Corporate taxpersonal tax on equity - personal
  tax (on interest)
• In the case of perpetual debt, the present value
  of the tax shield is (we use as
  a discount factor)
• and
• where is the corporate tax rate, is the
  marginal rate of personal tax, and is
  effective tax rate on equity.

50
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
• Leverage now decreases value.

51
Capital Structure Does matterThe Cost of
Financial Distress
CF to Equity
CF to Taxes
CF to Debt
52
Capital Structure Does matterThe Cost of
Financial Distress

• 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 matterThe 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
• 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
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
Conflict of interestExample

• Consider a firm with 50 of 1-year debt.

BOOK
MARKET
56
Incentives for Costly Games
A) The shareholders of the firm favor risky
projects.
B) Shareholders favor high dividends. They gain
the full benefit, but decline in the firm
value is shared with bondholders.
57
Capital Structure Does matterThe Cost of
Financial Distress
CF to Equity
CF to Taxes
CF to Debt
Financial Distress Cost
58
Capital Structure Does matterThe Cost of
Financial Distress

• 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.

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
Summary
Maximum value of firm
Costs of financial distress
Market Value of The Firm
PV of interest tax shields
Value of levered firm
Value of unlevered firm
Optimal amount of debt
Debt