Lecture Notes on Corporate Finance Theory

Norvald Instefjord

University of Essex

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1 Introduction

1.1 These lecture notes contain material for a 30 hours master course on corporate finance theory. They

are based partly on the new book by Tirole: The Theory of Corporate Finance; and also on Dixit: The Art of

Smooth Pasting; and Ziegler: A Game Theory Analysis of Options, and in addition on a number of academic

journal articles.

1.2 Central to the theory of corporate governance is the separation of ownership and control: the individ-

uals who own the corporation are not the same as those who have control. Much of corporate finance theory

are based on problems that arise because of this separation: problems related to transparency, executive ac-

countability, governance failures, but there are also benefits to be had: corporate ownership and restructuring

can take place quickly and efficiently, and investors who may be considered “active” can quickly establish an

influential stake in a corporation to exert pressure on the board and the management.

1.3 Corporations own real assets that generate a productive cash flow. They finance their operations by

issuing financial claims which have claims on the productive cash flow of the corporations. There are two

claims that empirically have dominated this role: debt and equity claims. A debt claim is a fixed claim on

the firm’s cash flow – the repayment plan is contractually established. If the firm fails to service its debt the

firm either needs to restructure its debt liability – i.e. renegotiate its debt contract – or choose to default on

its debt liability – which involves a formal process of bankruptcy with possible liquidation of the firm’s assets.

Much of corporate finance theory has focussed on explaining the security design features of the financial claims

used for financing corporations.

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1.4 Finally, corporations issue liabilities on which they can choose to default. To study the characteristics

of strategic default models it is required that dynamic models are used, and this has opened the avenue for

contingent claims valuation techniques in corporate finance. The final part of the course looks at some simple

continuous time models of investment and strategic default.

2 Fisher Separation, Net Present Value, Modigliani-Miller, Miller, and Myers

2.1 The Fisher Separation Theorem states that the firm’s investment decision is independent of the

preferences of the owner, and that the investment decision is independent of the financing decision. This

theorem has an enormous impact on corporate finance, in that we can separate out investment decisions and

financing decisions, and that we do not need to take account of what type of owner the firm has when making

investment decisions (e.g. the firm does not have to invest with low risk because it has risk averse owners).

The future profit of the firm depends on it’s level of investment i:

π(i) = f(i)− (1 + r)i

where (1 + r)i is the cost of raising the investment cost i in the capital market. The first order conditions

give us

π′(i) = 0 = f ′(i)− (1 + r)

Therefore, the marginal project f ′(i) earns just the market rate of return – all other projects earn a rate

higher than the market rate of return. Why should all owners of the firm agree to this strategy? Consider

an owner with utility function u(x, y) where x is future payoff and y is current investment, and who does not

consider borrowing/lending in the capital market, i.e. x = f(i) and y = i. The indifference curve of this owner

can be described as

du =∂u

∂xdx +

∂u

∂ydy = 0

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or, by rearranging,

−∂u/∂y

∂u/∂x=

dx

dy=

df

di

It is clear that depending on the implied time value of money embedded in the utility function u, the optimal

investment for the owner may be such that dfdi is greater or smaller than 1 + r, i.e. does not necessarily have

to lie at the optimal point of investment above. However, if we allow the investor to borrow and lend risk free

and amount z, such that x = f(i) + z(1 + r) and y = i + z, we consider the indifference curve above. Now, if

the option to borrow or lend is used,

dx

dy= (1 + r)

In the special case that the option to borrow or lend is not used, dxdy = df

di = (1 + r), so in either case we are

on an indifference curve which is tangential to the capital market line 1 + r. Obviously there is an interest

in pushing this line furthest to the north-west, so this line is tangential to the investment opportunity set at

the point where dfdi = 1+ r. This point is agreed on irrespective of preferences and financing, and corresponds

with the optimal point f ′(i) = 1 + r we found above.

2.2 Fisher Separation gives rise to the concept of net present value, which is a measure of the value added

for any investment project. Since Fisher Separation implies that any real investment should be measured

against an equivalent market based investment, we can talk in unambiguous terms about the “value” of a

project as the expected discounted future cash flow of the project (where we use the appropriate market based

discount rates). The net present value of a project is the difference between the value of the project (dis-

counted future cash flow) and the cost of the project (the investment outlay), and is a measure of the “value

added” of the project. Whereas this concept is simple, it can be difficult to implement the net present value

rule in practice. Cash flows can be difficult to evaluate as we should really only include the incremental cash

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flow associated with the investment decision – sunk cost needs to be ignored and opportunity cost needs to be

incorporated appropriately. Often, for instance, the opportunity cost of “doing nothing” is not zero because

the corporation is in some way losing out and the alternative to investment is not very easy to get a handle

on. It is unclear how such “parameter risk” or “estimation risk” should be accounted for in the net present

value analysis. A second problem is that the net present value “decision rule”, which is invest if positive net

present value, almost never can be implemented in this simple form. There may be agreement about whether

to go ahead with an investment but disagreement about timing. In this case the opportunity cost of choosing

an alternative time for investment must be evaluated as an option value of delaying the decision. These option

values can be very difficult to evaluate and may need additional modeling. We look at some timing problems

at the end of these lecture notes.

2.3 To illustrate the capital structure puzzle of Modigliani and Miller, consider an arbitrage-free market

in which the firm’s debt and equity are traded. The fact the market is arbitrage-free, implies that we can

assign state prices ps to every state s, such that any claim issued in the market with state-contingent payoff

xs takes the value

qx =∑

s

psxs

The firm has a productive cash flow ys, and claims on that cash flow c1s, c

2s, . . . , c

ns such that, for every state

s, the claims consume fully the productive cash flow

ys = c1s + c2

s + · · ·+ cns

Using the state prices, we multiply and sum over all states

V =∑

s

psys =∑

s

ps(c1s + c2

s + · · ·+ cns ) =

∑s

psc1s +

∑s

psc2s + · · ·+

∑s

pscns = V 1 + V 2 + · · ·+ V n

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so that the total value of the firm is constant and independent of the way in which the cash flow is split into

its various liabilities. Or at the very least, this holds as long as there are no third party claimants on the firm’s

productive cash flow (such as taxes or bankruptcy costs), and the firm’s liability structure does not influence

the state prices of the economy. Whereas the second caveat is reasonable the first is not. The tax system

in most countries favor some specific capital or liability structure, and if the firm defaults on one of its debt

claims it will normally incur some deadweight cost of financial distress. This has given rise to the so called

“trade-off” theory of capital structure: The firm seeks to take on debt because of its tax benefits, but not too

much because this increases the likelihood of financial distress. The optimal capital structure balances the

two: the marginal tax benefits of debt equals the marginal expected bankruptcy costs at the point where the

capital structure is optimal.

2.4 The capital structure puzzle refers to the fact that firms spend so much time and resources on the

design of its liability structure when a first approximation should indicate it is pretty much irrelevant to the

total value of the firm. The answer must lie elsewhere, and the corporate finance theory has, therefore, looked

for answers to the corporate finance puzzle using frameworks that go beyond the simple trade off argument

applied above, where the two main extensions apply agency theory and the economics of information.

2.5 To see that taxes are not a straightforward explanation, consider the following argument put forward

by Miller (1977). Suppose the firm has earnings y every period (ignore risk), which are taxed at the corporate

rate τC before being distributed to claim holders. If distributed to equity holders as income to equity, there

is an additional tax of τE imposed. If distributed to debt holders as income on debt, the firm can deduct the

income as an expense so deducts the corporate tax of τC , but the holders of the debt must pay a private tax

on debt income τB. Suppose that debt is perpetual with face value D and coupon rate k, which means that

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every period the income paid to debt holders equals kD, and the retained earnings y − kD is paid to equity

holders. This means that the total cash flow after tax is

kD(1− τB) + (y − kD)(1− τC)(1− τE) = y(1− τC)(1− τE) + kD(1− τB)(

1− (1− τC)(1− τE)1− τB

)

Discounting the value using the period discount rate r, we find the value of a levered firm as

V L =y(1− τC)

r+

kD(1− τB)r

(1− (1− τC)(1− τE)

1− τB

)= V U + B

(1− (1− τC)(1− τE)

1− τB

)

where V U is the value of an unlevered firm and B is the market value of the firm’s debt. The idea in Miller

is that if there is undersupply of debt in the economy at large, the taxation of debt income is likely be lower

than the taxation of equity income, since it is the investors who pay the least tax on debt that are drawn fist

to the debt market. Therefore, the factor 1 − (1−τC)(1−τE)1−τB

is likely to be positive, and there is an aggregate

increase of debt to the market by firms seeking to exploit the tax advantage. Similarly, if there is oversupply

of debt, the taxation of debt income is likely to be greater than the taxation of equity income, and firms seek

to reduce their borrowing to take advantage of the tax penalty on debt. The only stable equilibrium is the

one where the tax advantage is zero, or when 1− (1−τC)(1−τE)1−τB

= 0, in which case

V L = V U

i.e. leverage is irrelevant. Note however that there is a non-trivial level of industry borrowing necessary to

achieve the equilibrium, yet at the equilibrium point the individual firms are indifferent between increasing

and decreasing their borrowing. Note also that it is essential that there is no short selling allowed, otherwise

the Miller model would not have any equilibrium. To see this, consider the equilibrium point above where the

marginal investor in the debt market has the same tax burden as the marginal investor in the equity market.

This must imply that there are active investors in the debt market who has a smaller tax burden than the

marginal investor – these would be willing to buy more debt by shorting equity – and similarly there will be

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active investors in the equity market who pay less tax on equity income than the marginal investor in the

equity market, these would be willing to buy more equity by shorting debt. Thus, the short selling constraint

is always binding, and it is impossible to find a stable equilibrium point when the short selling is lifted.

2.6 Myers (1977) model deals with the so called debt overhang problem. This problem arises when the

firm has an existing debt liability that dwarfs its current assets, at the same time as it considers to invest in

future growth opportunities. We suppose the value of the growth opportunities become known at some time

before the investment decision is made, so the efficient investment plan is given by the rule to invest whenever

y ≥ i

where y is the realization of the present value of the future growth opportunities and i is the investment

cost. Suppose the model ends immediately after the investment decision is made: then the payoff to the debt

holders is

min(B, y) if investment is made, and

0 if investment is not made

where B is the contractual payment on the debt liability. It is supposed the firm has zero existing assets. The

corresponding payoff to the equity holders is

y −min(B, y) if investment is made, and

0 if investment is not made

We notice that for the equity holders, there is an incentive to stick with the efficient investment plan only for

B = 0:

y ≥ i ⇔ (y −min(B, y)) ≥ i, and B = 0

Therefore, the existence of a debt liability B > 0 leads to inefficiencies in investment.

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2.7 The following example illustrates the efficiency. Suppose ex ante, y is distributed as

y ∼ U [1, 2]

and that i = 1.3. The debt liability B = 0.2. If investment is made ex post, the payoffs to the debt holders

are

0.2 if investment is made

0 otherwise

and the payoffs to the debt holders are

y − 0.2 if investment is made

0 otherwise

The efficient investment plan dictates that investment is made for y ≥ 1.3, therefore the value of debt is

V ∗B =

∫ 2

1.30.2dy = 0.2(2− 1.3) = 0.14

and the value of the equity, net of investment, is

V ∗E =

∫ 2

1.3(y − 0.2− 1.3)dy =

(1222 − 1.5(2)

)−

(121.32 − 1.5(1.3)

)= 0.105

The total firm value under efficient investment is, therefore, V ∗ = V ∗B +V ∗

E = 0.245. Now consider the optimal

investment plan for the equity holders. They invest optimally when y − 0.2 ≥ 1.3, or y ≥ 1.5. This yield a

debt value of

V ∗∗B =

∫ 2

1.50.2dy = 0.2(2− 1.5) = 0.1

and an equity value of

V ∗∗E =

∫ 2

1.5(y − 0.2− 1.3)dy =

(1222 − 1.5(2)

)−

(121.52 − 1.5(1.5)

)= 0.125

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The total value of the firm is, therefore, V ∗∗ = V ∗∗B + V ∗∗

E = 0.225. The value of equity has increased by 0.02

and the value of debt has decreased by 0.04. The net effect is a decrease in the total firm value of 0.02.

2.8 The debt overhang problem gives rise to a theory of renegotiation. Here we look at ex ante renegotiation

of the debt liability, i.e. before the present value y is known. Consider the above example with an arbitrary

debt liability of B. We can work out the value of the debt liability and the value of the equity as functions of

B:

V ∗∗B =

∫ 2

1.3+BBdy = B(2− 1.3−B) = 0.7B −B2

and

V ∗∗E =

∫ 2

1.3+B(y − 1.3−B)dy =

(1222 − (1.3 + B)2

)−

(12(1.3 + B)2 − (1.3 + B)2

)= 0.245− 0.7B + 0.5B2

We notice that the value of the debt liability has an interior optimum: if we differentiate V ∗∗B with respect to

B we find the optimal point

B∗∗ = 0.35

at which point the value of the debt reaches its maximum point

maxV ∗∗B = 0.06125

This implies that if the debt holders have a nominal debt liability of B = 0.4, they would increase their debt

value by accepting a new debt contract with a nominal debt liability of B′ = 0.35. Although the new debt

contract has a lower nominal value, its economic value is increased.

2.9 The Myers model also allows for ex post renegotiation, after the present value of the growth opportunity

is known. Since the equity holders investment policy is dictated by investment whenever

y −min(y, B) ≥ i

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there will be states where investment is not optimal initially, i.e. where y−min(y, B) < i, but where a volun-

tary debt renegotiation to a new lever B′ such that y −min(y,B′) ≥ i will make both parties better off. The

debt holders now receive a payoff of B′ ≥ 0 (since uncertainty is now resolved they receive this amount for

sure), and the equity holders receive a payoff of y −min(y, B′)− i ≥ 0.

2.10 If the debt holders can anticipate to renegotiate ex post, they will never accept a renegotiated set-

tlement ex ante that involves B′ < B, since this puts them in a worse bargaining situation ex post. If ex

post renegotiation is not possible and the debt holders have all bargaining power ex ante, the debt hold-

ers will renegotiate to the point B∗∗ if B > B∗∗, or not renegotiate at all otherwise. If the equity holders

have all bargaining power, they will renegotiate to the lowest debt liability B′ which leaves the debt holders’

wealth equally well off. Thus, if the debt holders’ value is V ∗∗B (B) and there exists a B′ < B such that

V ∗∗B (B′) = V ∗∗

B (B), the equity holders suggest the renegotiated debt B′ and the debt holders accept this

contract. This implies that if the equity holders have the bargaining power, they will always renegotiate the

debt to a level at which the expected inefficiency of investment is lower or the same, as would happen if the

debt holders have all the bargaining power.

2.11 Two factors prevent renegotiation, or make the renegotiation process costly. The first factor involves

coordination costs or free riding costs among several debt holders. If the debt structure involves a large debt

holder and a small debt holder, the small debt holder realizes that the large debt holder has stronger incentives

to renegotiate – so may refuse settlement. Thus, he can free ride on the large debt holder. The second factor

involves asymmetric information. The debt holders may be reluctant to accept a renegotiated settlement if

the equity holders have more information than they have. This is due to the fact that the equity holders may

seek to use their information advantage strategically to achieve renegotiation benefits also in states where

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renegotiation is not necessary in order to induce investment. To prevent the equity holders to pursue such

strategies, it may be optimal not to accept all proposed renegotiated contracts.

2.12 Exercises:

1. Suppose the corporate tax rate is 35 % flat and constant, and the private tax rate varies between 0 and

40% depending on the individual. We assume the corporate tax rate applies to all income distributed

from corporations as equity (equity is taxed at zero percent privately is the firms don’t pay dividends and

the investors don’t realize capital gains); and that the private tax rate applies to all income distributed

from corporations as debt. We assume that the annual corporate income (across a very large set of small

corporations) is 100, and that the firm distributes c as coupon payments on perpetual corporate debt

and 100 − c as the residual annual equity income. The discount rate is 10% after tax, and we assume

that the marginal private tax rate on debt income τ depends on c:

τ = 0.004c

such that for c = 0 there is zero private tax on debt income and for c = 100 there is 40% private tax on

debt income.

(a) What is the average capital structure in equilibrium? What is the value of all debt? And of all

equity? And debt-equity together?

(b) Suppose now that there is only a single firm in the economy. What is the optimal capital structure

for this firm? What is the value of all debt? And all equity? And debt-equity together? Explain

any discrepancy with (a).

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2. A firm has the following assets: cash worth 10 and an investment project that costs 100 and yields the

present value y which is uniformly distributed between 90 and 200: y ∼ U [90, 200]. Assume the firm

also has a debt liability of 50. The decision to invest can be deferred until the true present value y is

realized.

(a) Ignoring the debt liability, what is the optimal investment plan and the risk neutral value of the

firm ex ante?

(b) Now take into account the debt liability when working out the optimal investment plan for the

equity holders. Ignore renegotiation options. What is the risk neutral value of the firm ex ante

now?

(c) Finally work out the optimal investment plan for the equity holders taking into account ex ante

renegotiation (assume the debt holders have all the bargaining power) and ex post renegotiation

(assume the equity holders have all the bargaining power). What are the respective firm values ex

ante in these cases?

3 Agency Models

3.1 The original agency model in corporate finance is the one by Jensen and Meckling (1976). This model

involves a firm whose production function transfer inputs x into output in the form of financial benefit and

non-pecuniary benefits that can be consumed by the manager/entrepreneur. Let C(x) be the cost of the input

and P (x) the value of the financial benefits, and let

B(x) = P (x)− C(x)

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Assume x∗ is such that

x∗ = arg maxx

B(x)

Assume B(x∗) = 1. Now define the financial value of producing non-pecuniary benefits as

F = B(x∗)−B(x) = 1−B(x)

This can be interpreted as the cost of choosing an input vector x different from the one that maximizes the

financial value of the firm. Of course, it may be optimal to choose F > 0, and we shall see what a rational

manager/entrepreneur would do if he was the sole owner of the firm. Define a utility function U(F,B) such

that F + B = 1, which yields the Lagrangian maximization programme

maxF,B,λ

U(F, B)− λ(F + B − 1)

The first order conditions for utility maximization yield

∂U

∂F= λ

∂U

∂B= λ

F + B = 1

Pick an indifference curve for the manager/entrepreneur:

dU =∂U

∂FdF +

∂U

∂BdB = 0

which yields the relationship

∂U/∂F

∂U/∂B= −dB

dF= 1

where the last equality follows by applying the first order conditions above at the optimal point. Therefore,

in general the “corner solution” B = 1 and F = 0 is not the optimal one even when there is no outside owners

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of equity.

3.2 We shall now see what happens when the manager/entrepreneur sells 1 − α of his stake to outside

shareholders (let’s now call him just the manager). We assume the outside shareholders are rational enough

to realize what optimal actions the manager now takes, but do not have the power to influence his actions

even though they may be harmful to their claim. The manager’s Lagrangian now looks takes the form

maxF,B,λ

U(F, αB)− λ(F + B − 1)

where the objective function now takes into account that the manager owns only α percent of the equity and

is therefore entitled to only α percent of the financial benefits. The constraint takes into account, however,

that the outside shareholders realize that the firm needs to operate on the original constraint. The first order

conditions are

∂U

∂F= λ

∂U

∂B=

λ

α

F + B = 1

If we now investigate the indifference curve for the manager, we find that

∂U/∂F

∂U/∂B= −dB

dF= α < 1

Using a geometric argument, we find that the utility level of the manager has shifted downwards. Also, we

find that it must be the case that the financial value of the firm has decreased, while the cost of non-pecuniary

perks has increased. The total welfare has decreased, since the outside shareholders have zero utility from the

transaction (they get what they pay for), and the welfare effects are measured solely by the manager’s utility.

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The loss is due to agency costs of outside equity.

3.3 Jensen and Meckling also talk about agency costs of outside debt. This problem is sometimes known

as the risk shifting problem. A simple example under risk neutrality can illustrate this problem. Let the cash

flow of the firm be distributed such as to have a mean preserving property with binomial cash flow

x =

1 + ε with prob 12

1− ε with prob 12

The mean cash flow is E(x) = 1 and the variance of the cash flow is ε2. Now, if the firm has a debt contract

with face value 1, the cash flow of the debt, xB, is

xB =

1 with prob 12

1− ε with prob 12

and the cash flow of the equity, xE , is

xE = x− xB =

ε with prob 12

0 with prob 12

It is pretty obvious that from the equity holders point of view an increase in risk (in terms of the variance of

the cash flow) is desirable as this enables the equity holders to “expropriate” cash flow from the debt holders.

We will get back to the risk shifting problem under the sections where we look at continuous time models of

debt financing.

3.4 The risk shifting problem appears to be costly to the debt holders, but given that the debt holders

can anticipate the risk shifting problem at the stage where debt is issued the real costs (if any) are borne by

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the equity holders. A further example illustrates this point. Suppose the firm has a cash flow x1 or x2, where

x1 =

1 + ε with prob 12

1− ε with prob 12

or

x2 =

1 + 10ε− µ with prob 12

1− 10ε− µ with prob 12

where ε and µ are fixed constants. Suppose 0 < ε < µ < 9ε. In a risk neutral world, the equity holders would

benefit from choosing the low risk cash flow x1 if no debt is issued, as E(x1) = 1 > 1− µ = E(x2). The risk

obviously does not matter here. If a debt contract is issued with face value 1, we find that the cash flow to

the debt holders is either

xB1 =

1 with prob 12

1− ε with prob 12

or

xB2 =

1 with prob 12

1− 10ε− µ with prob 12

and the corresponding cash flow to the equity holders is either

xE1 =

ε with prob 12

0 with prob 12

or

xE2 =

10ε− µ with prob 12

0 with prob 12

Since µ < 9ε, it follows that ε < 10ε− µ, so the equity holders would benefit more from the high risk option.

But this means that the total value of the firm is 1− µ, and the debt contract is initially worth 1− 5ε− 0.5µ

and the equity contract is worth 5ε− 0.5µ. Therefore, by selling the firm to rational debt holders who realize

that risk shifting will occur, the equity holders retain an equity value of 5ε−0.5µ and raises proceeds from the

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debt issue of 1− 5ε− 0.5µ. This is less than the value when no debt is issued. Therefore, the costs of the risk

shifting problem is really borne by the equity holders. The equity holders will therefore benefit from issuing

debt with covenants which prevent risk shifting, and this typically occurs in practice as corporate debt is

issued under the caveat that should significant changes occur to the assets of the firm the debt is immediately

redeemable.

3.5 Exercises:

1. Consider an entrepreneur with cash A and an investment opportunity that costs i > A. The project

has return R with probability p and 0 with probability 1 − p. The success probability p is pH if the

entrepreneur exerts managerial effort and pL = pH −∆ if he shirks. The entrepreneur receives private

benefits B if he shirks and nothing otherwise. The entrepreneur sells the project to outside investors

and retains a return Rb ≤ R in the case the project succeeds.

(a) Work out the incentive compatibility constraint guaranteeing that the entrepreneur exerts effort.

(b) Work out the maximum pledgeable return to outside investors, and also the minimum level of cash

A needed to secure external financing.

(c) Would it matter if the outside investors had the option of giving an incentive contract to induce

effort in this model? Explain.

4 Asymmetric Information

4.1 This section deals with corporate financing under asymmetric information which has become a large

part of corporate finance. The section is largely based on chapter 6 in Tirole’s book. The literature that deals

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with issuing securities under asymmetric information assumes that there is a entreprenuer/manager who has

superior information issuing a claim to a market with less sophisticated information. There is no problem

with this assumption when we are talking about newly started companies or companies who have not yet

issued external claims. However, most issues take place by companies who have already issued claims. This

obviously raises some questions about whether information that is private within the firm at the time where

external claims have been issued also may leak to some market participants when issuing further claims. These

issues are largely ignored and the basic entrepreneur/manager model is often assumed. An exception is the

Myers and Majluf model in which a corporation employs a manager who seeks to maximize the value of its

current equity holders. It takes therefore the view that management is in control of the issue but rarely take

advantage of this to pursue own objectives – i.e. the manager internalizes the welfare of the firm’s shareholders.

However, when seasoned equity issues takes place there is an obvious question of whether the manager should

internalize the welfare of its existing shareholders or its new shareholders. This is analogous to the situation

where we consider current action to combat pollution or climate change that has long-term implications for

future generations – but the future generations have no direct say in the decision making process. It is up to

us to decide how much we should internalize the future generations’ welfare in our decision. Often, moreover,

there is a conflict between current and future investors. The existing models largely incorporate a manager

who internalizes the current shareholder’s welfare at the cost of future investors’ welfare. This results in

situations where “optimal” long term decisions may be foregone in order to capture “sub-optimal” short term

gains. The fact that information is released through financing choices makes the “optimality” condition am-

biguous: it is not clear whether we should maximize with respect to full information or partial information sets.

4.2 To illustrate the problem of capital transfers in an asymmetric information model, consider the fol-

lowing simple risk neutral model. Suppose an entrepreneur owns a project but has no funds to pay for the

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investment cost i himself. The project has a return of R in the case of success and a return of 0 in the case

of failure. There are two types of projects, good or bad. A good project has success probability p such that

pR > i and a bad project has success probability q such that i > qR (negative NPV) or qR ≥ i but pR > qR

(positive NPV). The capital market is competitive and requires a return of 0. The manager knows his own

project’s type. The market has a probability distribution of project types – a project is good with probability

α and bad with probability 1− α. Define

m = αp + (1− α)q

Suppose the entrepreneur tries to sell his project by promising a return R − Rb to the investors. They are

willing to put up the investment if their expected return is at least as great as the investment cost i. If they

could observe project type perfectly, they would require

p(R−RGb ) = i

if the project is of good type and

q(R−RBb ) = i

if the project is of bad type. Clearly, if qR < i there is no solution to the latter equation, and if qR ≥ i the

solution is such that RBb < RG

b . If they cannot observe project type perfectly, and supposing all project would

seek financing, the constraint becomes

m(R−Rb) = i

If mR < i, there can be no solution to this equation, and all funding ceases to happen. This case is feasible only

if qR < i (there are some negative NPV projects seeking funding). Good projects are therefore underfunded

by the sheer existence of bad projects.

20

If mR ≥ i there will be funding, but in this case the return promised to investors is such that investors

break even on average, i.e.

m(R−Rb) = i

Investors make money on the good type and lose money on the bad type projects, i.e.

p(R−Rb) > i > q(R−Rb)

This is a case of over investment, where the good projects cross subsidize the bad projects.

4.3 The basic model can be applied to a study of market timing in which firms issue shares when stock

prices are “high”. There may be several reasons for such market timing but one is that adverse selection is

less of an issue during boom times. Consider, in addition to the terms defined above, an observable variable τ

which can be both positive or negative, such that the success probability is p + τ for good projects and q + τ

for bad projects. The condition for financing becomes now

(α(p + τ) + (1− α)(q + τ))R ≥ i

or

(m + τ)R ≥ i

The better the market conditions, the more likely is it that firms can obtain financing despite the problem

of asymmetric information, and the more likely it is that the capital market switches from a state of under

investment to a state of over investment.

4.4 Now consider the situation that the firm initially has assets in place. The entrepreneur has already

invested in a project which, with success probability p or q (depending on good or bad type, respectively), will

21

yield a return of R, and otherwise nothing. Assume the probability of good and bad types are α and 1− α,

respectively, as before. If the market values the firm at average values, therefore, the firm is undervalued if of

good type and overvalued if of bad type.

We assume the entrepreneur initially owns all assets. The entrepreneur may seek to make an additional

investment on top of the existing assets at a cost i. The asset has return R with success probability τ , and

we assume

τR > i

such that the investment opportunity is profitable both for the good and the bad type. This implies that the

success probability for a good type, conditional on new investment, is p + τ , and the success probability for

a bad type, conditional on new investment, is q + τ . The existing assets cannot be liquidated to raise the

investment cost i, and any new claims issued must have a claim on the project “portfolio” of the existing

asset and the new project – i.e. no claim can be issued against the new project alone.

Consider a pooling equilibrium where investment takes place regardless. In this case the return offered to

the investors RI must be such that

(α(p + τ) + (1− α)(q + τ))RI = i

or just

(m + τ)RI = i

where m is defined as above. Although this condition ensures new financing, a problem arises with respect

to an entrepreneur managing a good type firm. The initial claim is pR, but there is no guarantee that the

retained share in the firm exceeds this amount, i.e. there is no guarantee that

(p + τ)(R−RI) ≥ pR

22

which can be rearranged as

τR ≥ p + τ

m + τi

If the “average” success probability m is close to p there is unlikely to be a problem, since the condition then

reduces to τR > i which was assumed initially. However, if m is significantly smaller than p the new project

must be very good in order to guarantee the entrepreneur in charge of the good type firm wishes to invest. In

this case we get a separating equilibrium in which only the bad type seeks to issue stock for new investment.

The return promised to the new investors is RBI such that

(q + τ)RBI = i

The implication of a separating equilibrium is that there is a negative stock price response when new equity

is issued. The pre-announcement price of equity is

V0 = α(pR) + (1− α)((q + τ)R− i)

The post-announcement price of equity is

V1 = (q + τ)R− i

Since pR > (q + τ)R− i (this follows from the fact that pR > (p + τ)(R−RBI ) = (p + τ)(R− i/(q + τ))), the

stock price announcement is negative when a separating equilibrium is played:

V1 − V0 < 0

Conversely, when a pooling equilibrium is played the stock price reaction is zero.

4.5 The problem of adverse selection when the firm has assets in place when seeking new investment leads

to the so called pecking order hypothesis of financing. This hypothesis states that financing sources can be

23

ranked according to their “information intensity”. Internal sources of financing, e.g. retained earnings, have

no information intensity as they do not involve outside uninformed parties. Debt may be issued externally, but

will normally have lower information intensity than equity, as the debt value is less sensitive to the private

information signal of the entrepreneur, and hence influence the issuing process to less degree than equity.

Debt issues will, therefore, normally be ranked ahead of equity issues. Then finally, as a last resort, the firm

issues equity, which is highly information intense. We will show the effect of issuing default free debt in our

example. This requires that we “re-jig” our model somewhat since as long as we operate with returns of R or

0, there is no distinction between debt and equity. Suppose there is a salvage value RF , such that conditional

on success the project has return R + RF = RS , and otherwise a return of RF . If

mRS + (1−m)RF > i

risky debt can be issued to raise the investment cost i, and if

RF ≥ i

even risk free debt can be used. It is easy to confirm that the use of risk free debt carries no cost of financing

(as there is no informational cost associated with issuing a claim whose value is independent of information).

4.6 Exercises:

1. A firm owns assets with random value x and new investment opportunities which cost 100 with random

present value y. The firm has no cash reserves. Suppose x and y are independent bivariate random

variables:

x =

100 with prob 12

200 with prob 12

y =

105 with prob 12

115 with prob 12

24

Assume the manager of the firm has private knowledge of the realizations of x and y before making the

investment decisions whereas the investors are uninformed.

(a) What is the state by state economically efficient investment plan? What is the ex ante value of the

firm assuming investment efficiency?

(b) Suppose the manager issues new equity to finance the investment cost. What is the investment

plan that maximizes the value of the current shareholders? What is the ex ante value of the firm

assuming this investment plan is adhered to?

(c) Suppose the manager issues new debt instead. Explain what happens now. What is the ex ante

value of the firm in this instance?

(d) A debt overhang is an existing debt liability that matures after the investment decision is made.

Explain what impact a debt overhang would have on your answer in (c)?

5 Control Rights, Takeovers and Security Design

5.1 The Aghion-Bolton model is about allocation of control rights between insiders (entrepreneur/manager)

and outsiders (investors/shareholders). The idea is simple – by transferring control rights to outsiders, the

insider can increase his pledgeable income to outsiders, and thus facilitate access to funding for profitable

projects. Formally, we can illustrate the idea in a simple model. An entrepreneur with wealth A considers a

project with cost i. The outside financing need is the shortfall i− A. The project succeeds with probability

p and yields a return of R. We assume p = pH if the entrepreneur behaves and p = pL < pH if the entrepre-

neur misbehaves. In the latter case the entrepreneur consumes private benefits B. This is the moral hazard

problem. Also assume that there is the possibility of taking interim action such that firstly, the probability

25

of success is increased uniformly by τ > 0 (i.e. pH + τ or pL + τ depending on the entrepreneur’s behavior),

and secondly, implies a private cost γ > 0 to the entrepreneur. Motivating examples include: switch to a

more profitable strategy; severing a long-time relationship with a collaborator; downsizing; divesting a “pet”

division. The interim action determines welfare in this model.

We notice that the incentive compatibility constraint for the entrepreneur is independent of the interim

action (as τ increases the success probability uniformly). Let Rb be the entrepreneur’s share of success return

R. If the entrepreneur behaves, the return is (pH + τ)Rb, and if the entrepreneur misbehaves, the return is

(pL + τ)Rb + B. Therefore, to guarantee that the entrepreneur behaves it is necessary that

(pH + τ)Rb ≥ (pL + τ)Rb + B Rb ≥ B

pH − pL=

B

∆p

which is independent of τ and would also arise if the success probabilities were pH and pL.

The profit-enhancing action may reduce aggregate welfare (this is the interesting case) if

τR < γ

because in this case the increase in expected profits is not enough to offset the cost γ. From a welfare point

of view, this action should not be undertaken. Suppose control rights are allocated to investors. Since the

investors do not incur the cost γ (which is borne by the entrepreneur) they would nonetheless like the action

undertaken as it enhances profits. Therefore, the net return to outside investors is

(pH + τ)(R−Rb) = (pH + τ)(

R− B

∆p

)

and the NPV of the total investment is (which is also equal to the entrepreneur’s utility as the entrepreneur

can issue outside claim at competitive prices)

NPV = (pH + τ)R− I − γ

26

If we in contrast assume that control rights are not relinquished to outsiders, there is no profit enhancing

action taken at the interim stage, so the return to outside investors is

pH(R−Rb) = pH

(R− B

∆p

)

and the NPV of total investment

NPV = pHR− I > (pH + τ)R− I − γ

Relinquishing control to outsiders will, therefore, decrease the NPV and the entrepreneur’s utility by γ− τR,

but on the other hand increase pledgeable income by τ(R− B

∆p

). If the entrepreneur is not cash constrained

he will always prefer not to relinquish control and implement the first best in this case. The problem comes

when the entrepreneur is cash constrained and

pH

(R− B

∆p

)< i−A < (pH + τ)

(R− B

∆p

)

In this case, it is necessary that the entrepreneur relinquishes control in order to start the project in the

first place. The argument that projects requiring substantial investment up front may never be implemented

unless the entrepreneur increases pledgeable income to investors is an argument in favor of shareholder value.

5.2 The Grossman-Hart model deals with value-enhancing takeovers. We consider a firm with value v

which may be taken over by a raider to increase the value to v = v + 1 (synergy gains normalized to 1). The

market value is assumed informationally efficient prior to the takeover, i.e. the value before a bid is announced

is v, and we assume the raider bids a price v + P with P being the takeover premium. We assume there

is a continuum of shareholders with mass 1, which implies there is no pivotal shareholder who can “tip the

balance” in the case a bid is put forth. We assume the raider needs to acquire at least κ ∈ (0, 1) to gain control

(i.e. to be able to realize the synergy gains). We also define β as the probability of success of an unrestricted

27

and unconditional offer (defined as an offer that is open to all shareholders willing to tender). The main

result in Grossman-Hart is that regardless of the synergy gains being positive the success probability is always

limited to

β = P

The reason is the following. If β > P then each shareholder would prefer to hold on to his shares, yielding

βv + (1− β)v = v + β(v − v) = v + β

than to tender his shares, yielding only

v + P

A reverse argument can be made for β < P . Exactly κ percent of the shares in the firm must be tendered

(if more are tendered β = 1 and if less β = 0 - so each shareholder must tender with probability κ – this can

be derived formally starting from a finite group of shareholders and taking the limit as the number goes to

infinity but we will ignore these details here). Given this equilibrium in the takeover market, we find that the

raider’s profits are

π = κ(β − P ) = 0

This is called the free rider problem of takeovers: takeovers fail with positive probability because the non-

tendering shareholders are “free riding” on the tendering ones such as to receive a “home made” takeover

price of v + 1 which always dominate the actual takeover price v + P .

5.3 The case which permits value-decreasing takeovers has implications for security design. This is the

Grossman-Hart “one-share/one-vote” result. Consider the post-takeover value v and the pre-takeover value

v, defined as above, and private benefits of control w to the raider. For v < v it is a weakly dominant strategy

28

for all shareholders to tender provided the takeover premium P defined as above is greater than zero: P ≥ 0.

Similarly, when P ≤ v − v it is a weakly dominant strategy not to tender. However, consider the range

v − v < P < 0

The shareholders are better off collectively if the takeover fails for sure than if it succeeds for sure, since P < 0.

Moreover, each shareholder’s incentive to tender increases if it is more likely that the others tender also, as

the synergy gains are negative (the opposite of the free riding effect in the case of value enhancing takeovers).

We can, therefore, imagine two types of equilibria: the “trust equilibrium” where each shareholder does not

tender trusting the others not to tender also; and the “suspicion equilibrium” where each shareholder tenders

believing the others to tender also. The trust equilibrium gives the maximum payoff to all shareholders and

the suspicion equilibrium the minimum payoff. The two equilibria may co-exist, but the trust equilibrium

Pareto dominates (from the shareholders’ point of view) the suspicion equilibrium. Moreover, the suspicion

equilibrium would disappear if a friendly arbitrageur were to appear to outbid the raider. In sum, therefore,

the raider is likely to be constrained to bid P ≥ 0, but it is feasible the raider can take control with P = 0.

The issue in question is the distribution of ownership rights. We can imagine two structures: either

the company has a uniform distribution of the shareholders’ voting power so that the voting power of each

shareholder is directly proportional to the number of shares he owns. This is the “one vote/one share”

ownership structure; or the company has a dual class share structure, with A shares carrying voting power

and the B shares carrying no voting power. If the raider needs to acquire κ of the votes, it is necessary that

he buys at least a ≥ κ of the company in the first case, and at least a ≥ κ of the A shares in the second case.

If the A shares have a financial claim on β of the company, the net surplus for the raider is

w − a(v − v)

29

in the first case and

w − aβ(v − v)

in the second case. Since β < 1

w − a(v − v) < w − aβ(v − v)

in the second case. The implication is that the raider is more likely to take over the firm with a dual class

share structure, and the financial value of the company is more likely to be less (because of the negative

synergy gains). Therefore, the shareholders are less likely to pay the same price for the B shares with a dual

class structure as they are for the ordinary shares under a “one vote/one share” structure. The A shares are

on the other hand likely to have the same price as the ordinary shares under a “one vote/one share” structure

(since the premium P is zero). Therefore, it is a value-maximizing strategy for the shareholders that the “one

vote/one share” structure is adopted at the outset.

5.4 Exercises:

6 Timing of Investment, Strategic Default Models of Corporate Debt

6.1 The last section deals with continuous time models applied to corporate finance. Continuous time

models have the advantage that they are fully dynamic models, so any aspect of corporate finance studied

will automatically be evaluated from a dynamic point of view. The workhorse model is based on geometric

Brownian motions,

dy

y= µdt + σdZ

where dyy is the instantaneous percentage change in the state variable y (the return), µdt is the deterministic

part (where µ is constant – µ is instantaneous return), and σdZ is the stochastic part (the noise – where σ

30

is constant and equal to the standard deviation of returns – and where dZ is the increments of a standard

Brownian motion). The increments of Z: Zt+s − Zt, have a distribution which is independent of t, and is

normal with mean zero and variance proportional to the length of the time interval: N(0, s). In this section

we assume risk neutrality – or at the very least a risk neutral uncertainty structure.

6.2 When working with Brownian motions the standard rules of calculus don’t apply as the sample paths

are nowhere differentiable. This is due to the fact that even for small time increments the noise term don’t

vanish. The rule of calculus are instead given by Ito’s lemma:

df(y) =∂f

∂ydy +

12

∂2f

∂y2(dy)2 = fy(µydt + σydZ) +

12fyy(dy)2

where

(dy)2 = σ2y2(dZ)2 = σ2y2dt

where we use the rule that dt2 = dtdZ = 0 and dZ2 = dt. Any function of y must evolve according to Ito’s

lemma. If we are considering a function of time as well as y: f(y, t), Ito’s lemma states that the rules of

calculus are “normal” for the time variable but must include the second order term for the Brownian motion

y:

df = fy(µydt + σydZ) + ftdt +12fyyσ

2y2dt

6.3 Taking a geometric Brownian motion y we can derive an absolute Brownian motion x by applying

Ito’s lemma to the function x = ln y:

dx =1ydy +

12

(− 1

y2

)σ2y2dt =

(µ− 1

2σ2

)dt + σdZ

31

This Brownian motion can, in contrast to the geometric Brownian motion, go negative. Applying Ito’s lemma

again to the absolute Brownian motion x using the function y = ex we recover the orignial geometric Brownian

motion:

dy = exdx + ex(dx)2 = µydt + σydZ

The expected discounted value of an absolute Brownian motion x with drift rate µ− 12σ2 is equal to

E0

∫ ∞

0e−rtxtdt =

∫ ∞

0e−rtE(xt)dt

We know that the expectation of the dZ term is zero, hence the expectation is the expected change in x:

E(xt) = x0 +(

µ− 12σ2

)t

Hence,∫ ∞

0e−rt

(x0 +

(µ− 1

2σ2

)t

)dt =

∫ ∞

0e−rtx0dt +

∫ ∞

0e−rt

(µ− 1

2σ2

)t dt

The first term is∫ ∞

0e−rtx0dt =

(−1

re−rtx0t

)∞

0

=x0

r

and the second, using integration by parts,

∫ ∞

0e−rt

(µ− 1

2σ2

)t dt =

((µ− 1

2σ2

)t

(−1

r

)e−rt

)∞

0

−∫ ∞

0

(µ− 1

2σ2

)(−1

r

)e−rtdt

Evaluating the first term on the right hand side we find it vanishes, so it suffices to work out the second:

∫ ∞

0e−rt

(µ− 1

2σ2

)t dt =

((µ− 1

2σ2

)(− 1

r2

)e−rt

)∞

0

=µ− 1

2σ2

r2

Putting it all together, we find

E0

∫ ∞

0e−rtxtdt =

x0

r+

µ− 12σ2

r2

32

6.4 Discounting a geometric Brownian motion y with drift parameter µ and volatility parameter σ we use

the associated absolute Brownian motion x. Since yt = ext we know that the associated random variable xt

is normal N(x0 +

(µ− 1

2σ2)t, σ2t

)with density function

f(xt) =1√2πs

e−�

xt−m√2s

�2

dxt

Here we use s = σ√

t and m = x0 +(µ− 1

2σ2)t. Thus, we can work out the expected value

E(yt) =∫ ∞

−∞extf(xt)dxt

where we use the density function above. We find

E(yt) =∫ ∞

−∞

1√2πs

exte−�

xt−m√2s

�2

dxt

=∫ ∞

−∞

1√2πs

ext−meme−�

xt−m√2s

�2

dxt

= em

∫ ∞

−∞

1√2πs

ezte−�

zt√2s

�2dzt

dxtdxt (zt = xt −m)

= em

∫ ∞

−∞

1√2πs

ezt− z2t

2s2 dzt

= eme12s2

∫ ∞

−∞

1√2πs

e−�

zt−s2√2s

�2

dzt

= em+ 12s2

where the integral disappears (i.e. equals 1) by the fact that the integrand is a density function of a normal

random variable with mean s2 and variance s2. Using the definitions of m and s, we find

E(yt) = ex0+µt

Therefore, the discounted value of a geometric Brownian motion is

E

(∫ ∞

0yte

−rtdt

)=

∫ ∞

0e−rt+x0+µtdt =

y0

r − µ(y0 = ex0)

33

6.5 We can also work out the present value of a geometric Brownian motion using a differential equation

approach. Consider an underlying state variable x which is an absolute Brownian motion

dx =(

µ− 12σ2

)dt + σdZ = νdt + σdZ

The earnings flow is a geometric Brownian motion ex with drift parameter µ and volatility parameter σ. Let

F (x) be the present value of all future earnings ex. By Ito’s lemma we know that

dF = Fxdx +12Fxx(dx)2 =

(νFx +

12σ2Fxx

)dt + σFxdZ

Investors with the right to the earnings flow get E(dF + exdt), i.e. the expected change in the value of the

earnings flow plus the current earnings. The return they require is rFdt, i.e. the market return on the current

value. This yields a differential equation E(dF + exdt) = rFdt which can we written as

12Fxxσ2 + νFx − rF + ex = 0

Guess on F = eλx for the homogenous part. This yields Fx = λF and Fxx = λ2F , so the homogenous equation

reduces to

12σ2λ2 + νλ− r = 0

which has two roots λ1 and λ2 (one is negative and the other positive). Therefore, the general solution to the

homogenous part is

FH = Aeλ1x + Beλ2

with A and B arbitrary constants (since the ODE is linear the linear combinations of any two linearly

independent solutions span the entire solution set). For the particular solution we need to find any solution:

34

Guess F = Kex to get Fx = F and Fxx = F , which implies

12σ2K + νK − rK + 1 = 0

which has solution

K =1

r − (ν + 1

2σ2) =

1r − µ

The general solution is, therefore,

F = Aeλ1x + Beλ2x +ex

r − µ

To identify the specific solution, we can use the following trick. Suppose the current earnings are ex. Now

imagine that we simply increase the starting point from ex to ex+∆ = exe∆. This should increase the value F

by a factor of e∆:

F (x + ∆) = e∆F (x)

Using the general solution, we find

Aeλ1(x+∆) + Beλ2(x+∆) +ex+∆

r − µ= e∆

(Aeλ1x + Beλ2x +

ex

r − µ

)

which necessitates that A = B = 0 unless λ1 or λ2 equal one. The λ parameter cannot however be equal to

one since this would imply that

12σ2λ2 + νλ− r

∣∣∣∣λ=1

=12σ2 + ν − r =

12σ2 + µ− 1

2σ2 − r = µ− r 6= 0

Therefore, F (x) = ex

r−µ .

6.6 We now consider the problem of “harvesting” an earnings flow y which follows a geometric Brownian

motion with drift parameter µ and volatility parameter σ, at some optimal point in time at a fixed one-off

35

cost of K. This is a so called “optimal stopping problem”. We know that at the point of harvest y∗, the value

of the harvest is the discounted expected value of the future earnings flow

F (y∗) =y∗

r − µ

Before the point of harvest, the value is a function G(y), which by Ito’s lemma evolves according to the

stochastic differential equation

dG = Gydy +12Gyy(dy)2 =

(Gyµy +

12Gyyσ

2y2

)dt + GyσydZ

Since this claim is an asset its return is the required market return E(dG) = rGdt, which yields the differential

equation

12Gyyσ

2y2 + Gyµy − rG = 0

If we guess a solution G = yλ, we find Gy = λyλ−1 and Gyy = λ(λ− 1)yλ−2. Putting these into the ODE we

find

12λ(λ− 1)σ2yλ + λµyλ − ryλ = 0

where we can eliminate yλ and identify the two roots λ1 and λ2 which obtains:

λ1,2 =− (

µ− 12σ2

)±√(

µ− 12σ2

)2 + 2σ2r

σ2

one positive and one negative. The general solution to the ODE above is, therefore

G(y) = Ayλ1 + Byλ2

where A and B are arbitrary constants. Suppose λ1 < 0 < λ2. If y ↓ 0, we know that there will be no harvest

in the foreseeable future, hence G(0) = 0. However, since λ1 < 0, limy↓0 yλ1 = ∞. Therefore, the coefficient

A must be zero. Moreover, at the point of harvest there must be value matching:

G(y∗) = F (y∗)−K

36

or

Byλ2 =y∗

r − µ−K

This condition is however not sufficient to determine the missing parameters B and y∗ so we need one more

condition. This is the optimality condition, the so called “smooth pasting condition”, which states that G

must not only match the value at y∗ it must also be tangential:

dG(y)dy

∣∣∣∣y=y∗

=dF (y)−K

dy

∣∣∣∣y=y∗

or

Bλ2yλ2−1 =

1r − µ

If we multiply by y on both sides and divide by λ2, we find

Byλ2 =y

r − µ−K =

1λ2

y

r − µ

where the first equality follows from the value matching condition and the second from the smooth pasting

condition. This enables us to identify the optimal stopping time (optimal time of harvest):

y∗ = K(r − µ)(

1− 1λ2

)−1

= K(r − µ)λ2

λ2 − 1

and the coefficient B:

B =y∗

r − µ

1λ2

(1y∗

)λ2

=(

y∗

r − µ−K

)(1y∗

)λ2

The value of the investment opportunity G is, therefore,

G(y) =(

y∗

r − µ−K

)(y

y∗

)λ2

The right hand side here has a natural interpretation. The first term is the value conditional on harvest being

made, and the second term is a number between zero and one which can be interpreted as the risk neutral

37

probability of making the harvest (notice that λ2 is not just positive, it is also greater than one). You can see

that as y → y∗ the second term approaches one.

6.7 Consider an example: the earnings flow follows a geometric Brownian motion with drift parameter

0.03 and volatility parameter 0.5 – i.e. we are expecting an earnings growth of 3% per year (continuously

compounded) and a volatility of 50% per year. The risk free interest rate is 5% per year (continuously

compounded) and the investment cost is 100. If we take the break even level for investment we get

y

0.05− 0.03− 100 = 0

which yields y = 2. We know that the optimal trigger is

y∗ = 100(0.05− 0.03)λ2

λ2 − 1

where

λ2 =− (

0.03− 120.52

)+

√(0.03− 1

20.52)2 + 2(0.5)20.05

0.52≈ 1.12

so the investment trigger

y∗ = 21.120.12

≈ 19

This is much higher than intuition tells us, as the optimal value at investment is 190.05−0.03 ≈ 950, i.e. 9.5

times the investment cost. The property that strategic decision making involves a higher degree of delay than

intuition tells us is something we will come back to when looking at strategic default on debt liability.

6.8 Consider a firm which has made investments that generate an earnings flow y which follows a geometric

Brownian motion with drift parameter µ and volatility parameter σ. The firm has also incurred a debt liability

which involves paying a coupon flow c to the debt holders. Assume the debt is perpetual. If the debt is risk

38

free, the discounted value of the coupon flow is

BF =∫ ∞

0e−rtcdt =

(− c

re−rt

)∞0

=c

r

Since the debt contract is a corporate contract there is a default risk, so cr is the upper bound on the value

of the debt. We assume the equity holders can time the default such as to maximize the value of the equity.

This implies that in states where y < c but the equity holders nonetheless chooses NOT to default they must

support the firm by an injection of new equity. The “dividend” flow y − c may, therefore, be positive or

negative depending on the level of current earnings. Let the debt contract have a value B. By Ito’s lemma,

the evolution of B is

dB = Bydy +12Byy(dy)2 + cdt =

(Byµy +

12Byyσ

2y2 + c

)dt + ByσydZ

where the term cdt comes from the fact that there is a continuous coupon flow received by the debt holders.

The required return on the debt is rBdt, such that putting E(dB) = rBdt yields the differential equation

12Byyσ

2y2 + Byµy − rB + c = 0

The homogenous part has the solution

BH = A1yλ1 + A2y

λ2

where A1, A2 are arbitrary constants and λ1, 2 are as in section 6.6. For the particular solution we guess on

B = D constant, which implies By = Byy = 0 and

−rD + c = 0

so the particular solution is just the value of the risk free debt BF . Therefore, the value of the corporate debt

contract is

B = A1yλ1 + A2y

λ2 +c

r

39

Now if y ↑ ∞, there is no chance of default in the foreseeable future. The value of the debt contract must

therefore B → cr , i.e. the value of risk free debt. Since λ1 < 0 and λ2 > 0, the first term vanishes but the

second explodes, therefore A2 = 0. The value of corporate debt is, therefore (using A = A1)

B = Ayλ1 +c

r

The total value of the firm is just the discounted value of the earnings flow:

V =∫ ∞

0e−rtytdt =

y0

r − µ

and the value of the equity is the residual value after subtracting the value of the debt (use y = y)):

E = V −B =y

r − µ−Ayλ1 − c

r

The coefficient A can be determined by considering the optimal stopping time for default y∗∗. First, at default

the value of equity must be zero (value matching):

y

r − µ−Ayλ1 − c

r

∣∣∣∣y=y∗∗

= 0

Second, at default the value of the equity must be tangential to zero (smooth pasting):

1r − µ

−Aλ1yλ1−1

∣∣∣∣y=y∗∗

= 0

Combining the two, we find

Ayλ1 =y

r − µ− c

r=

y

r − µ

1λ1

where the first equality comes from the value matching condition and the second from the smooth pasting

condition. This yields the optimal stopping time for default

y∗∗ =c

r(r − µ)

λ1

λ1 − 1

40

The coefficient A can now be identified. Using the value matching condition above we find

A =(

y∗∗

r − µ− c

r

)(1

y∗∗

)λ1

and the value of the corporate debt contract prior to default is equal to

B =(

y∗∗

r − µ

)(y

y∗∗

)λ1

+c

r

(1−

(y

y∗∗

)λ1)

Here, the value of the debt contract has an intuitive interpretation. The term(

yy∗∗

)λ1

can be interpreted

as the risk neutral probability of default, so the first term is the expected payoff to the debt holders in the

case of default (as in this case the debt holders can collect the default value y∗∗r−µ), and the second term is the

expected payoff to the debt holders in the case of no-default (as in this case the debt holders can collect the

risk free debt value cr ).

6.9 We now consider the case that there are deadweight default costs associated with the option to default.

If the firm defaults at an earnings level y, the debt holders receive

(1− α)y

r − µ

and there is a deadweight loss associated with default equal to

αy

r − µ

The debt contract must have the same dynamic prior to default as before, so we can write

B = A1yλ1 +

c

r

The deadweight loss (let’s call it L) can be thought of as an asset in the same way as the debt contract, so it

must satisfy (prior to default) the ODE

12Lyyσ

2y2 + Lyµy − rL = 0

41

and will have the general solution

L = A2yλ1 + A3y

λ2

with A2 and A3 arbitrary constants. As y ↑ ∞, there is no chance of default in the foreseeable future so L ↓ 0,

implying that A3 = 0. Therefore, the default costs are L = A2yλ1 . The equity claim is the residual

E = V −B − L =y

r − µ−A1y

λ1 − c

r−A2y

λ1 =y

r − µ− (A1 + A2)yλ1 − c

r

where A1 + A2 is to be decided. We notice that the strategic default decision for the equity holders is exactly

the same now as it was with α = 0, so

A1 + A2 =(

y∗∗

r − µ− c

r

) (1

y∗∗

)λ1

implying the same default trigger

y∗∗ =c

r(r − µ)

λ1

λ1 − 1

This is intuitive. As the equity holders don’t bear any of the default costs ex post, they don’t take α into

account when they choose when to default. The debt holders lose out ex post, however, as the value of the

debt just prior to default is

B = A1yλ1 +

c

r= (1− α)

y

r − µ(y = y∗∗)

implying

A1 =(

(1− α)y∗∗

r − µ− c

r

)(y∗∗)−λ1

so

B = (1− α)y∗∗

r − µ

(y

y∗∗

)λ1

+c

r

(1−

(y

y∗∗

)λ1)

Similarly, the value of the default costs can be derived through the value matching condition

L = A2yλ1 = α

y

r − µ(y = y∗∗)

42

implying

A2 = αy∗∗

r − µ(y∗∗)−λ1

so the value of the default costs prior to default is

L = αy∗∗

r − µ

(y

y∗∗

)−λ1

6.9 Exercises:

1. Consider an earnings flow that follows a geometric Brownian motion

dy

y= µdt + σdw

where µ and σ are constants, dt denotes time increments and dw denotes the increments of a standard

Brownian motion.

(a) Calculate the expected discounted earnings flow given by

E0

(∫ ∞

0e−rtytdt

).

(b) Suppose the earnings flow y belongs to an investment project which has not yet been started. The

investment cost needed to initiate the project is a fixed lump sum K. Derive the optimal investment

trigger y∗.

(c) Suppose the investment is partially financed by perpetual debt with a fixed coupon flow c. Suppose

the equity holders can decide strategically when to default on this debt contract. What is the

optimal default trigger y∗∗?

43

Answers to Exercises:

Section 2:

1. (a) We know that for all c such that τ < 0.35 there is an incentive to issue more debt and for all c such

that τ > 0.35 there is an incentive to issue less. Hence, the equilibrium is described by τ = 0.35 or

0.004c = 0.35 which implies c = 350/4 = 87.5. The average capital structure is, therefore, 87.5%

debt and 12.5% equity. The individual firm is, however, indifferent between issuing debt and equity

so some firms may be more levered than the average and others may be less.

(b) If there is only one firm present, the firm no longer acts as a price taker since it realizes that

changes to its capital structure will lead to changes in the private tax rate on debt. Hence, it seeks

to maximize

maxc

(100− c)0.650.1

+c(1− τ)

0.1=

(100− c)0.65 + c(1− 0.004c)0.1

It suffices to maximize the numerator, so we get the first order condition

−0.65 + 1− 0.008c = 0

or,

c =0.350.008

=3508

= 43.75

44