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(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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 2 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 3 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 4 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 5 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 6 / 45
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 7 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 8 / 45
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 9 / 45
Weak irrelevance result: Modigliani–Miller (1958)
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 10 / 45
Weak irrelevance result: Modigliani–Miller (1958)
What about if no agent has shares of both firms?
Proof.
. . .
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 11 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 12 / 45
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)]
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 13 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 14 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 15 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 16 / 45
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}
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 17 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 18 / 45
Investors’ budget sets
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 19 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 20 / 45
Stock market Equilibrium
(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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 21 / 45
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`)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 22 / 45
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`)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 23 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 24 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 25 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 26 / 45
Limited liability and bankruptcy
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 27 / 45
Limited liability and bankruptcy
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}
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 28 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 29 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 30 / 45
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 31 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 32 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 33 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 34 / 45
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]
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 35 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 36 / 45
Investors’ budget sets
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 ...
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 37 / 45
Investors’ budget sets
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 38 / 45
Investors’ budget sets
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 39 / 45
Stock market Equilibrium
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 40 / 45
Stock market Equilibrium
(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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 41 / 45
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)
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 42 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 43 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 44 / 45
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
V. F. Martins-da-Rocha (FGV) Corporate Finance 3 trimestre, 2012 45 / 45