Lecture 8: Two period corporate debt model
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
A two-period model with investment
At time 1, the firm buys capital k, using equity issuance s and
debt: the firm issues a debt with face value b, at unit price
q(k, b), hence the budget constraint:
χq(k, b)b + s = k
The parameter χ > 1 reflects the tax shield effect - interest
expenses are subsidisized. For each dollar of debt issued, the
firm receives a subsidy χ − 1 .
At time 2, the firm produces, and obtains a profit π = zk α , where
z is an idiosyncratic shock, distributed according
R ∞to a cumulative
distribution function F, with mean 1 : E(z) = 0 sf (s)ds = 1.
Default decision
The firm will default if its profits are not large enough to repay
its debt, i.e. if z < z ∗ , with
z ∗ k α = b.
If the firm does default, equity holders get nothing, while
bondholders share the firm profits, net of bankruptcy costs.
The parameter 0 < θ < 1 determines the share of profits which
are recovered.
Debt and Equity Value
Debt value is given by a standard pricing equation, assuming risk
neutrality and a discount factor β :
!
Z ∞
Z z∗
zk α
q(k, b) = β
dF (z) .
dF (z) +
θ
b
z∗
0
The firm equity value is
Z
∞
V =β
z∗
(zk α − b) dF (z).
Maximization
The firm chooses k, b, s, z ∗ to maximize its present discounted
value
Z ∞
α
max∗ β
(zk − b) dH(z) − k + χq(k, b)b
k,b,z
z∗
subject to:
Z
∞
q(k, b) = β
Z
dF (z) +
z∗
∗ α
z k
z∗
0
!
zk α
dF (z) ,
θ
b
= b.
The firms solves
Z
max
β
∗
k,z
+ χβ
∞
k α (z − z ∗ ) dF (z) − k
z∗
z ∗ kα
Z
∞
Z
dF (z) +
z∗
0
!
z∗
θzk α dF (z)
Optimality
This program can be solved by writing the two first order
conditions, with respect to k and z ∗ , which determine the
optimal investment and financing. First, we have
(Z
Z
∞
1 = βαk
α−1
∗
(z − z ) dF (z) + χ z
z∗
∗
∞
dF (z) + θ
z∗
which can be rearranged, given that E(z) = 1, as:
(
1 = βαk α−1
Z
1 + (χ − 1) z ∗ (1 − F (z ∗ )) + (θχ − 1)
z∗
zdF (
0
Z
)
z∗
zdF (z) .
0
Interpretation: note that if χ = θ = 1, we have the frictionless
limit: 1 = βαk α−1 E(z), the usual MPK = user cost condition.
When θ < 1, χ > 1, this is not true anymore, and the tax shield
χ and the bankruptcy cost θ both tend to decrease the user cost.
Optimality
The second first-order condition, with respect to z ∗ , yields, after
rearrangement
((1 − F (z ∗ )) (χ − 1) = χ (1 − θ) z ∗ f (z ∗ ) .
This equation determines z ∗ , and hence the optimal leverage
(increasing z ∗ means increasing the leverage and the probability
of default).
LHS: benefit of higher leverage – the tax shield conditional on
no default.
RHS: The cost of higher leverage is an increase in bankruptcy
costs through a higher probability of default - the change in
probability of default is f (z ∗ ) at the margin, and the default costs
are proportional to z.
zf (z)
If z lognormal then z 1−F
(z) is increasing, which guarantees a
unique solution to this equation.
Some additional intuition
We can follow BGG notation. Let
Z
Z
∗
∗
Γ(z ) = z
dF (z) +
z∗
z ∗ dF (z)
0
and
∗
z∗
Z
µG(z ) = µ
z∗
z ∗ dF (z)
0
denote bankruptcy costs where µ = 1 − θ.
Then the equity share is
1 − Γ(z ∗ ) + (χ − 1) (Γ(z ∗ ) − µG(z ∗ ))
The lender’s share is
Γ(z ∗ ) − µG(z ∗ )
Firm Value
Firm value is
Q(z ∗ )βk α − k
where
Q(z ∗ ) = 1 − Γ(z ∗ ) + χ [Γ(z ∗ ) − µG(z ∗ )]
zf (z)
1−F (z)
is increasing implies that the term in brackets is strictly
concave with an interior maximum at some z ∗∗ . So neither debt
or equity claimants desire z ∗ > z ∗∗ .
Optimal leverage
Optimality then implies choosing leverage so that the marginal
value of an additional unit of debt is equal to the marginal value
of an additional unit of equity:
χ Γ0 (z ∗ ) − µG0 (z ∗ ) = Γ0 (z ∗ )
This occurs at an interior value 0 < z ∗ < z ∗∗ . Since Q(0) = 1
and Q(z ∗∗ ) < 1 we have Q(z ∗ ) > 1. As long as χ > 1 issuing
debt raises firm value. Otherwise it would be optimal to set
z ∗ = 0 to avoid bankruptcy costs.
Optimal choice of k
The FOC then imply:
Q(z ∗ )αk α−1 = 1/β
which means that capital is above the efficient level owing to the
tax subsidy on debt.
We can interpret (βQ(z ∗ ))−1 as the user cost of capital.
Comparative Statics
The value of z ∗ is determined by the parameters χ, θ, and the
distribution F.
The comparative statics w.r.t. χ and θ is clear.
With regard to F , an interesting comparative static is a change in
the payoff distribution F. If F becomes more risky, at least for
most distributions, we will have less debt and less capital.
Intuitively, the debt contract becomes more inefficient, making
the cost of the debt higher. (Note: this is ex-ante higher risk, not
ex-post: this is different from risk-shifting)
Financial Frictions
Now assume that equity is fixed at some predetermined value n
(Net worth) and set χ = 1.
The firm borrows some initial amount qb = k − n
We can write the problem as:
Z ∞
α
max∗ β
(zk − b) dH(z) − k + n + q(k, b)b
k,b,z
z∗
subject to:
Z
∞
q(k, b) = β
dF (z) +
z∗
∗ α
z k
Z
= b.
q(k, b)b = k − n
0
z∗
!
zk α
θ
dF (z) ,
b
Maximization
Max problem
Z
max
β
∗
k,z
+ λ β
∞
k α (z − z ∗ ) dF (z) − k
z∗
∗ α
Z
∞
z k
Z
dF (z) +
z∗
!
z∗
α
θzk dF (z)
!
− (k − n) .
0
We can write this as
maxk,z ∗ {βQ(z ∗ )k α − λ(k − n)}
where
Q(z ∗ ) = 1 − Γ(z ∗ ) + λ (Γ(z ∗ ) − µG(z ∗ ))
Optimality
FOC:
Qβαk α−1 = λ
λ Γ0 (z ∗ ) − µG0 (z ∗ ) = Γ0 (z ∗ )
β (Γ(z ∗ ) − µG(z ∗ )) k α = k − n
Entrepreneur’s Investment Opportunity
Entrepreneur starts period with net worth N .
Entrepreneur borrows:
B = QK − N,
Q = price of capital (exogenous)
Project payoff:
ωRk QK α ,
0<α<1
Rk = aggregate (gross) rate of return on capital (exogenous)
ω = idiosyncratic shock to project’s return
2
2
Assume: ln ω ∼ N ( −σ
2 , σ ) ⇒ E[ω] = 1
Information Structure
No asymmetric information ex ante:
Rk is known to both lender and entrepreneur before investment
decision.
ω is realized after investment decision.
Asymmetric information ex post:
ω is freely observed by entrepreneur.
To observe ω, lender must pay
µωRk QK α
Parameter 0 ≤ µ < 1 measures the cost of monitoring and hence
the magnitude of credit market frictions.
Debt Contract
Entrepreneur and lender agree to a standard debt contract (SDC)
that pays lender an amount D as long as bankruptcy does not
occur.
If ωRk QK α ≥ D:
Entrepreneur pays D to lender and keeps residual profits.
If ωRk QK α < D:
Entrepreneur declares bankruptcy and gets nothing.
Lender pays bankruptcy cost to monitor entrepreneur and keeps
profits net of bankruptcy cost.
Payoffs to Entrepreneur and Lender
Bankruptcy occurs if ω ≤ ω:
ω≡
D
Rk QK α
Expected payoff to entrepreneur:
Z ∞
Z
k
α
ωR QK dΦ(ω) − ω
ω
∞
Rk QK α dΦ(ω)
ω
Expected payoff to lender:
Z
(1 − µ)
ω
k
α
Z
∞
ωR QK dΦ(ω) + ω
0
Rk QK α dΦ(ω)
ω
Competitive loan market: Lender must earn an expected (gross)
rate of return R on the loan amount B.
Payoffs as a Share of Expected Profits (Rk QK)
Define:
ω
Z
Γ(ω) ≡
Z
∞
ωdΦ(ω) + ω
0
dΦ(ω)
ω
Z
µG (ω) ≡ µ
ω
ωdΦ(ω)
0
Entrepreneur’s expected share of profits:
1 − Γ(ω)
Lender’s expected share of profits:
Γ(ω) − µG(ω)
Optimal Contract
Choose K and ω to solve:
max [1 − Γ(ω)] Rk QK α
K,ω
subject to the lender’s participation constraint:
[Γ(ω) − µG(ω)] Rk QK α = R(QK − N )
Lagrangean:
n
o
max [(1 − Γ(ω)) + λ (Γ(ω) − µG(ω))] Rk QK α − λR(QK − N )
K,ω
λ = Lagrange multiplier on the lender’s participation constraint
and hence measures the shadow value of an extra unit of net
worth to the entrepreneur.
The term in brackets reflects total firm value when valued using
the shadow price of external funds.
Optimality Conditions
FOC w.r.t. ω̄:
λ=
Γ0 (ω)
≥1
[Γ0 (ω) − µG0 (ω)]
FOC w.r.t. K:
α [(1 − Γ(ω)) + λ (Γ(ω) − µG(ω))] Rk QK α−1 = λRQ
FOC w.r.t. λ:
[Γ(ω) − µG(ω)] Rk QK α = R(QK − N )
External Finance Premium
FOCs imply:
αRk QK α−1 = ρ(ω̄)RQ
ρ(ω̄) =
h
λ
[1−Γ(ω)]+λ[Γ(ω)−µG(ω)]
i
≥1
ρ(ω̄) = external finance premium (EFP)
EFP is increasing in the default barrier ω:
ρ0 (ω̄) > 1
Leverage and Default:
The default barrier ω̄ is increasing in leverage:
QK
ψ(ω̄)
=
N
1 − (1 − α)ψ(ω̄)
where
ψ(ω̄) =
λ [Γ(ω) − µG(ω)]
1+
≥1
1 − Γ(ω)
ψ 0 (ω̄) > 0
Intuition:
An increase in leverage requires a higher default barrier to
increase the payoff to the lender relative to the entrepreneur.
The increase in the default barrier also implies a higher shadow
value of external funds λ.
An increase in net worth reduces the default barrier and lowers
the premium on external funds.
Example: Constant Returns to Scale (α = 1)
The default barrier is determined by the rate of return on capital
relative to the risk-free rate of return:
Rk
= ρ(ω̄)
R
Given ω̄, capital expenditures are determined by available net
worth:
QK
= ψ(ω̄)
N
Combining these, we obtain a positive relationship between the
premium on external funds and leverage:
Rk
QK
=s
, s0 > 0
R
N