(Some theoretical aspects of) Corporate Finance

(Some theoretical aspects of)Corporate Finance

V. Filipe Martins-da-Rocha

Escola de Pos-graduacao em Economia

Fundacao Getulio Vargas

Part 1.b. Stock Market Equilibrium: Capital Structure of the Firm

V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 1 / 45

Outline: the financial structure of the firm

Irrelevance results: the Modigliani–Miller’s theorems

Taxation

Limited liability and bankruptcy

Irrelevance of the payout policy


The financial structure of the firm: literature

Modigliani, F. and Miller, M.The cost of capital, corporation finance, and the theory ofinvestment.American Economic Review 48, 1958

Stiglitz, J.A re-examination of the Modigliani–Miller theorem.American Economic Review 59, 1969

Stiglitz, J.On the irrelevance of corporation financial policy.American Economic Review 64, 1974


The financial structure of the firm: literature

DeMarzo, P. M.An extension of the Modigliani–Miller theorem to stochasticeconomies with incomplete markets and interdependent securities.Journal of Economic Theory 45, 1988

DeMarzo, P. M. and Duffie, D.Corporate financial hedging with proprietary information.Journal of Economic Theory 53, 1991

Purnanandam, A.Financial distress and corporate risk management: Theory andevidenceJournal of Financial Economics 87, 2008


The financial structure of the firm

From now on, we fix a production plan y and we focus on therelevance of the financial plan β

Assume the objective of the manager is to maximize the value

V = E +D + y0

Recall thatI the (present value of the) debt is given by D = q · βI the equity price E (and the security price q) is determined at

equilibrium and may depend on β

The manager maximizes the function β 7→ E(β) + q · βAssume for simplicity that y0 = 0

There is a natural question: Is there an optimal choice β??

The answer is: the financial policy of the firm is irrelevant


A simple case

Consider that there is a riskless asset a0 promising 1, i.e.,

∀s ∈ S, κs(a0) = 1

We denote by r0 the risk free interest rate, i.e.,

q(a0) =1

1 + r0

Assume that the firm can only issue (sell) the riskless bond a0, i.e.,

∃Z > 0, β = Z1a0

In other words, the firm commits to deliver at t = 1 the amount Zindependent of the state of nature


A simple case

The value of the firm is equity plus debt, i.e.,

V = E +D where D =1

1 + r0Z

the firm is said levered if D 6= 0

the firm is said unlevered if D = 0

the debt-equity ratio is defined by

D

E

a specific choice of β leads to a specific debt equity rationD(β)/E(β)


The conventional approach before 1958

Assume the firm starts without debt

the dividend of the firm δ1 = (ys)s∈S is risky

debt is not risky, issuing debt may attract investors and increasethe firm’s value

I no debt: investors can buy the risky cash flow (ys)s∈SI with debt: investors can still buy a risky cash flow (ys − Z)s∈S but

now they can also buy a riskless cash flow Z1S

a low level of debt reduces the contingencies where the firm will beunable to meet its obligations

however, for a high level of debt, the possibility for bankruptcybecomes non-negligible and the benefits of leverage (issuingnon-risky asset) disappears

it seems that there is an optimal level of debt


Weak irrelevance result: Modigliani–Miller (1958)

Theorem

Consider a stock market equilibrium{(y(j), β(j))j∈J , (q, E), (ci, θi, αi)i∈I

}Assume that

the two firms j1 and j2 have the same production plan, i.e.,y(j1) = y(j2)

but may have different financial policies β(j1) 6= β(j2)

Then the two firms have the same value, i.e.,

E(j1) + q · β(j1) = E(j2) + q · β(j2)



Proof.Assume there exists an agent i having shares of both firms, i.e.,

αi(j1) > 0 and αi(j2) > 0

Then there exists a family of state price deflators (λs)s∈S in RS++ such that

q =∑s∈S

λsκs

and∀j ∈ {j1, j2}, E(j) =

∑s∈S

λsδs(j)

This implies that

∀j ∈ {j1, j2}, V (j) ≡ E(j) + q · β(j) =∑s∈S

λsys



What about if no agent has shares of both firms?

Proof.

. . .


Strong irrelevance result: Stiglitz (1969, 1974)

Theorem

Consider a stock market equilibrium{(y(j), β(j))j∈J , (q, E), (ci, θi, αi)i∈I

}For any other financial plan (β(j))j∈J there exists another stockmarket equilibrium such that firms’ values remain unchanged


Strong irrelevance result: Stiglitz (1969, 1974)

More precisely, the following family{(y(j), β(j))j∈J , (q, E), (ci, θi, αi)i∈I

}is a stock market equilibrium where

E(j) = E(j) + q · [β(j)− β(j)]

θi = θi +∑j∈J

αi(j)[β(j)− β(j)]


Conditions for the validity of irrelevance results

Frictionless capital markets:I no transaction costsI no institutional restrictions on security trades

No taxes

Investors and firms have access to the same security (bond)markets

Unlimited liability of shareholders

Investor do not try to infer information about the likelihood ofstates of nature by observing firm’s financial policy


Conditions for the validity of irrelevance results

Linear effect: the change in the return on equity, induced by amodification of the firm’s capital structure, is a linear combinationof the returns on the assets traded by the firm

Consider there are derivative securities (like options) whose payoffis a function (possibly non-linear) of the future value of anotherasset

When the firm’s capital structure changes, not only the dividendof the stock changes but also that of the derivative securitieswritten on the firm’s shares

In general, the firm’s valuation will not be independent of thefirm’s capital structure

Gottardi, P.An analysis of the conditions for the validity of Modigliani–Millertheorem with incomplete markets.Economic Theory 5, 1995


The role of taxes

Taxes are paid at t = 1

We do not explain (model) the endogenous formation of tax rates

We do not explain the purpose of taxes (intermediaries, publicprojects)

Three tax rates

corporate tax rate tc on corporate income δ1 = y1 − κ1 · βpersonal tax rate ts on equity income

personal tax rate tb on financial assets (bonds) income

Assumption

all tax payers face the same tax rates

tax rates are constant and independent of the level of income towhich they apply


Assumptions

Since taxes apply to income, we assume that

every security a is claim to a non-negative amount, i.e.,

∀a ∈ A, ∀s ∈ S, κs(a) > 0

every firm j never takes the risk to be bankrupt, i.e., theproduction-finance plan (y, β) belong to D where

D ≡{

(y, β) ∈ Y × RA : ∀s ∈ S, ys > κs · β}


Investors’ budget sets

Each investor i takes as given:

each firm j’s production-finance plan (y(j), β(j))

the security price vector q and the equity price vector E

and chooses

a consumption plan (c0, c1) ∈ R+ × RS+a security portfolio θ ∈ RA+ and shareholdings α ∈ RJ+

satisfying budget restrictions



solvency restriction at t = 0

c0 + q · θ + E · α 6 ωi0 + (E + δ0) · ξi (1)

solvency restriction at every s ∈ S at t = 1

cs 6 ωis + (1− τb)κs · θ + (1− τs)(1− τc)δs · α (2)

The set of actions (c, θ, α) satisfying (1) and (2) is denoted byBi(q, E, τ)


Stock market Equilibrium

A stock market equilibrium with taxes is a list{(y(j), β(j))j∈J , (q, E), (ci, θi, αi)i∈I

}of

production-finance plan (y(j), β(j)) for each firm j

security and equity prices (q, E)

action (ci, θi, αi) of each investor i

such that



(a) investor i’s action is optimal, i.e.,

(ci, θi, αi) ∈ argmax{U i(c) : c ∈ Bi(q, E, τ)}

(b) security and stock markets clear, i.e.,

∀a ∈ A,∑j∈J

β(j, a) =∑i∈I

θi(a) and ∀j ∈ J,∑i∈I

αi(j) = 1

(c) commodity markets clear, i.e.,

cp +∑i∈I

ci =∑i∈I

ωi +∑j∈J

ys(j)

where cp0 = 0 and

cps ≡∑j∈J{τc + (1− τc)τs} ys(j) + {τb − τc − (1− τc)τs}κs · β(j)


Gains from corporate leverage

Assume that there exist two firms ju and j` such that

both firms have the same production plan, i.e., there existsy ∈ Y (ju) ∩ Y (j`) such that

y = y(ju) = y(j`)

firm ju is unlevered, i.e., β(ju) = 0

firm j` is levered, i.e., β(j`) > 0

firm j` is never bankrupt, i.e.,

∀s ∈ S, ys > κs · β(j`)


Gains from corporate leverage

Proposition

Assume that there is a stock market equilibrium{(y(j), β(j))j∈J , (q, E), (ci, θi, αi)i∈I

}Then

V (j`) = V (ju) +

[1− (1− τc)(1− τs)

1− τb

]D(j`)


Role of corporate taxes

Assume that personal income taxes coincide, i.e.,

τs = τb

thenV (j`) = V (ju) + τcD(j)

Firms have an incentive to raise capital via bond issues until thebankruptcy limit

Does not fit with data


Role of personal tax system

Assume that taxes matter, i.e.,

(1− τc)(1− τs) 6= 1− τb

then

if(1− τc)(1− τs) < 1− τb

then firms still have an incentive to raise capital via bond issuesuntil the bankruptcy limit

if(1− τc)(1− τs) > 1− τb

then firms have an incentive to fully finance their capital throughequity

these predictions do not fit with data


Do taxes may provide an optimal debt-to-equity ratio?

Theory

Miller, M. H.Debt and taxes.Journal of Finance 32, 1977

Empirical Literature

Mackie-Mason, Jeffrey K.Do taxes affect corporate financing decisions?Journal of Finance 45, 1990

Desai, M. A., Foley, C. F., Hines Jr., J. R.A multinational perspective on capital structure choice andinternal capital markets.Journal of Finance 59, 2004



Assume that only the riskless bond is available, i.e., A = {a0}Fix a firm and its production-finance plan (y, Z)

It may be the case that for some state s, we have

δs < 0 or equivalently ys < Z

In order to protect (and attract) investors, shareholders havelimited liability in the sense that a new shareholder at t = 1 is notcalled upon to make payments when δs < 0



If δs < 0 the firm is said to be bankrupt:I all the firm’s production ys is used to pay bondholdersI shareholders receive nothing

It is as if the firm is issuing a bond which yields

∀s ∈ S, bs = min{ys, Z}

One share of the firm yields

∀s ∈ S, δs = max{δs, 0} = max{ys − Z, 0}


What are the markets available to investors

Irrelevance of the financial policy

Every investor can purchase or sell any bond issued by firms

Possible relevance of the financial policy1 Investors can purchase or sell only a “pooled bond” based on

firms’ bonds

2 Investors can only purchase bonds issued by firms


Empirical Literature

Smith, C. and Warner, J.On financial contracting: an analysis of bond covenants.Journal of Financial Economics 7, 1979

Weiss, L. A.Bankruptcy resolution: Direct costs and violation of priority ofclaims.Journal of Financial Economics 27, 1990

Opler, T. C. and Titman, S.Financial distress and corporate performance.Journal of Finance 49, 1994


Firms objectives

We focused our attention on the effects of the firm’s decisions onthe assets’ rates of return

Other factors may guide the firms’ financial decisions. The capitalstructure may be viewed as

I a signaling device: Ross (1977), Leland and Pyle (1977)I a way of reducing agency costs: Jensen and Meckling (1976)I a way to allocate control rights between various claimants: Aghion

and Bolton (1988), Harris and Raviv (1989)


Irrelevance of the payout policy: A simple multi-periodmodel

A multi period stock market: t ∈ {0, 1, . . . , T}There is no uncertainty at every period

Bonds (or financial assets) are long-lived and pay dividends atevery period t

There is only one firm


Payout policies

The firm chooses:

An investment-production plan (x, y) ∈ Z, where

x = (x0, x1, . . . , xT−1) ∈ RT+

andy = (y1, . . . , yT ) ∈ RT+

xt represents the units of the good invested (inputs) at period t

yt+1 represents the units of the good produced (outputs) at periodt+ 1

The technological restrictions are represented by the set Z


Payout policies

The firm chooses:

A financial plan β = (β0, β1, . . . , βT−1) where

βt ∈ RA

A dividend policy n = (n0, n1, . . . , nT−1) where nt represents thenumber of shares at

I the end of period t, after payment of dividends nt−1δtI beginning of period t+ 1 before payments of dividends ntδt+1


Interpretation and notations

Denote by pt the ex-dividend price of one stock of firm j at period tafter announcing the payout policy

If nt > nt−1 then firm j is issuing shares and obtains the revenue

pt[nt − nt−1]

If nt < nt−1 then firm j repurchases shares and spends

−pt[nt − nt−1]


Interpretation and notations

Consider the firm’s production and payout policy

(x, y, β, n)

each investor i chooses the stockholding αt > 0 in units of stocksof the firm

one stock of the firm purchased at date t is a claim to the dividendδt+1 paid at period t+ 1

ntδt+1 = yt+1 − xt+1 − κt+1βt + pt+1[nt+1 − nt] + qt+1[βt+1 − βt]

for notational convenience, we let

y0 = 0, xT = 0, n−1 = 1 and β−1 = 0



Each investor i takes as given:

the firm’s production-investment plan (y, x) and payout policy(β, n)

the securities’ prices q = (qt)t and the stock prices p = (pt)t

and chooses

a consumption plan c = (ct)t ∈ RT+1+

a security portfolio θ ∈ RA×T and a stock-holdings vector α ∈ RT+satisfying ...



initially at t = 0

c0 + q0 · θ0 + p0 · α0 6 ωi0 + (p0 + δ0) · ξi (3)

for each t ∈ {1, . . . , T − 1},

ct + qt · θt + pt · αt 6 ωit + (κt + qt) · θt−1 + (δt + pt) · αt (4)

and finallycT 6 ωiT + κT · θT−1 + δT · αT−1

The set of actions (c, θ, α) satisfying the budget constraints is denotedby Bi(q, p)



initially at t = 0

c0 + q0 · (θ0 − ξi) + p0 · (α0 − 0) 6 ωi0 + δ0 · ξi (5)

for each t ∈ {1, . . . , T − 1},

ct + qt · (θt − θt−1) + pt · (αt − αt−1) 6 ωit + κt · θt−1 + δt · αt−1 (6)

and finallycT 6 ωiT + κT · θT−1 + δT · αT−1

The set of actions (c, θ, α) satisfying the budget constraints is denotedby Bi(q, p)



Given the firm’s capital budgeting and payout policy

(x, y, β, n)

a stock market equilibrium is a list of{(q, p), (ci, θi, αi)i∈I

}of

security and share prices (q, p)

action (ci, θi, αi) of each investor i

such that



(a) investor i’s action is optimal, i.e.,

(ci, θi, αi) ∈ argmax{U i(c) : (c, θ, α) ∈ Bi(q, p)}

(b) security and stock markets clear, i.e., for every 0 6 t < T − 1

∀a ∈ A, βt(a) =∑i∈I

θit(a) and∑i∈I

αit = nt

(c) commodity markets clear, i.e.,∑i∈I

ci = [y − x] +∑i∈I

ωi


No-arbitrage and state price deflator

The definition is the same apart from stock markets clearing condition:

∀t ∈ {0, . . . , T − 1},∑i∈I

αit = nt

The value Vt of the firm at period t is defined by

Vt = ptnt︸︷︷︸Et

+ qt · βt︸︷︷︸Dt

+(yt − xt)


Irrelevance of the payout policy

By no-arbitrage, there exists a vector of state prices

λ = (λ0, . . . , λT ) ∈ RT+1++

such that for every t ∈ {0, . . . , T − 1}

λtqt = λt+1(κt+1 + qt+1) and λtpt = λt+1(pt+1 + δt+1)

Observe that

λ0V0 =

T∑t=0

λt(yt − xt)

and

λtVt =

T∑τ=t

λτ (yτ − xτ )

The values of the firm are independent of the payout policy


How to explain the existence of an optimal capitalstructure?

Costs of agency

These costs are due to conflict of interests among agents involved inthe financial decisions:

debtholders, new shareholders, old shareholders and managers allenter into negotiations for different reasons

bringing them into agreement is costly, concessions are required toachieve at least a second-best solution


How to explain the existence of an optimal capitalstructure?

Costs of asymmetric information

Managers (insiders) may have a private information about theinvestment opportunities of the firm and related cash flows

the information asymmetry may lead to some inefficiencies in thefinancing decisions of the firms

managers may use their choice of capital structure as a crediblesignal to the market of their private information


(Some theoretical aspects of) Corporate Finance

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