[springer finance] contract theory in continuous-time models || an application to capital structure...

Chapter 7An Application to Capital Structure Problems:Optimal Financing of a Company

Abstract In this chapter we present an application to corporate finance: how tooptimally structure financing of a company (a project), in the presence of moralhazard. In the model the agent can misreport the firm earnings and transfer moneyto his own savings account, but there is an optimal contract under which the agentwill report truthfully, and will not save, but consume everything he is paid. Themodel leads to a relatively simple and realistic financing structure, consisting ofequity (dividends), long-term debt and a credit line. These instruments are used forfinancing the initial capital needed, as well as for the agent’s salary and coveringpossible operating losses. The agent is paid by a fixed fraction of dividends, whichhe has control over. At the optimum, the dividends are paid locally, when the agent’sexpected utility process hits a certain boundary point, or equivalently, after the creditline balance has been paid off. Because the agent receives enough in dividends,he has no incentives to misreport. When having a larger credit line is optimal, inorder to be allowed such credit, it may happen that the debt is negative, meaningthat the firm needs to maintain a margin account on which it receives less interestthan it pays on the credit line, as also sometimes happens in the real world. Thecontinuous-time framework and the associated mathematical tools enable one tocompute many comparative statics, by solving appropriate differential equations,and/or by computing appropriate expected values.

7.1 The Model

We follow mostly the approach and the model from DeMarzo and Sannikov (2006),while similar results have been obtained also by Biais et al. (2007).

Instead of assuming that the agent applies unobservable effort, we assume thatthe principal cannot observe the output process, and the agent may misreport itsvalue, and keep the difference for himself. We assume that the real output processis

dXut := μdt + vdB−u

t = (μ + utv)dt + vdBt , (7.1)

and the agent reports the process X:

dXt := (μ − utv)dt + vdB−ut = μdt + vdBt . (7.2)

J. Cvitanic, J. Zhang, Contract Theory in Continuous-Time Models, Springer Finance,DOI 10.1007/978-3-642-14200-0_7, © Springer-Verlag Berlin Heidelberg 2013

115

http://dx.doi.org/10.1007/978-3-642-14200-0_7

116 7 An Application to Capital Structure Problems: Optimal Financing

We assume μ,v are constants and, as before, u is FB -adapted.The agent receives a proportion 0 < λ ≤ 1 of the difference in the case of report-

ing a lower profit than is true, that is, in case ut ≥ 0. In principle, we could alsoallow him to over-report the change in cash flows, ut < 0, but it can be shown thatthis will not happen at the optimum. Thus, we assume

u ≥ 0. (7.3)

It is clear that this model is equivalent to the usual moral hazard model with un-observable effort, when the benefit of exhorting effort −utv is equal to additionalλutvdt in consumption.

The agent maintains a savings account S, unobserved by the principal, with thedynamics

dSt = ρStdt + λutvdt + dit − dct , S0− = 0. (7.4)

Here, ρ is the agent’s savings rate, dit ≥ 0 is his increase in income paid by theprincipal, and dct ≥ 0 is his increase in consumption. We assume c and i are rightcontinuous with left limits, hence so is S. It is required that

St ≥ 0, (7.5)

which is a constraint on the agent’s consumption c.The agent is hired until (a possibly random) time τ , which is specified in the

contract. It is assumed that he receives a payoff R ≥ 0 at the time of termination τ ,and that he is risk-neutral, maximizing

WA,c,u0− := W

A,c,u0 + �c0

where WA,c,u0 := W

A,τ,c,u0 := E−u

[∫ τ

0e−γ sdcs + e−γ τR

](7.6)

where �c0 is a possible initial jump in consumption. Here, we use the conventionthat ∫ t2

t1

·dct :=∫

(t1,t2]·dct .

Thus, the agent’s problem is

WA0− := sup

(c,u)∈A(τ,i)

WA,c,u0− = sup

(c,u)∈A(τ,i)

E−u

[∫[0,τ ]

e−γ sdcs + e−γ τR

], (7.7)

where the agent’s admissible control set A(τ, i) will be specified later.The principal’s controls are τ and i, which are F

X-adapted and thus, equiva-lently, FB -adapted. We only consider implementable contracts (τ, i), that is, suchthat there exists (c, u) ∈ A(τ, i) with WA

0− = WA,c,u0− . Similarly, the principal is also

risk-neutral, with expected utility

WP,τ,i,c,u0− := W

P,τ,i,c,u0 − �i0

:= E−u

[∫ τ

0e−rsd(Xs − is) + e−rτL

]− �i0. (7.8)

7.2 Agent’s Problem 117

We will specify the principal’s admissible set A later. The principal’s problem is

WP0− := sup

{W

P,τ,i,c,u0− : (τ, i) ∈A and (c, u) ∈ A(τ, i) is the agent’s

optimal control}. (7.9)

As usual in the literature, when the agent is indifferent between different actions,we assume he will choose one among those which are best for the principal. Weassume that the principal has to pay an amount K ≥ 0 in order to get the projectstarted. However, this does not change the principal’s problem (7.9).

We fix WA0− = wA and we want to find a contract which maximizes principal’s

expected utility, while delivering wA to the agent if he applies the strategy which isoptimal for the given contract.

We now specify some initial assumptions, with more being specified later.

Assumption 7.1.1

(i) γ > r ≥ ρ and rL + γR ≤ μ.(ii) Given (τ, i), each (c, u) in the agent’s admissible set A(τ, i) consists of a pair

of FB -adapted processes, c is non-decreasing and right continuous with left

limit, (7.3) and (7.5) hold, and Girsanov’s theorem holds for −u1[0,τ ].(iii) Each (τ, i) in the principal’s admissible set A consists of F

B -adaptedprocesses, i is non-decreasing and right continuous with left limit,E[(∫ τ

0 e−γ tdit )2] < ∞, and (τ, i) is implementable.

7.2 Agent’s Problem

We first study the agent’s problem (7.7). We start by showing that the agent willconsume everything he gets, and will keep his savings at zero. This is because therate at which the principal can save is higher or equal to the agent’s rate.

Proposition 7.2.1 Assume Assumption 7.1.1 holds. Given a contract (τ, i) ∈A, forany u, the agent’s optimal consumption is

dct = λutvdt + dit and thus S = 0. (7.10)

Proof Note that (7.4) leads to

d(e−ρtSt

) = λe−ρtutvdt + e−ρtdit − e−ρtdct .

For any c such that (7.5) holds, applying the integration by parts formula and thanksto the assumption that γ ≥ ρ, we have


WA,c,u0− = E−u

[∫[0,τ ]

e−(γ−ρ)se−ρsdcs + e−γ τR

]

= E−u

[∫[0,τ ]

e−(γ−ρ)s[λe−ρsusvds + e−ρsdis − d

(e−ρsSs

)] + e−γ τR

]

= E−u

[∫[0,τ ]

e−(γ−ρ)s[λe−ρsusvds + e−ρsdis

] + e−γ τR

− e−(γ−ρ)τ e−ρτ Sτ − (γ − ρ)

∫ τ

0e−(γ−ρ)se−ρsSsds

]

≤ E−u

[∫[0,τ ]

e−(γ−ρ)s[λe−ρsusvds + e−ρsdis

] + e−γ τR

],

and equality holds if and only if (7.10) holds. �

Thus, the agent’s problem can be rewritten as

WA0− = sup

u∈A(τ,i)

WA,u0− := sup

u∈A(τ,i)

E−u

[∫[0,τ ]

e−γ s[λusvds + dis] + e−γ τR

]. (7.11)

We next specify a technical condition on the agent’s admissible set A(τ, i).

Assumption 7.2.2 Given (τ, i), A(τ, i) is the set of FB -adapted process u ≥ 0 suchthat

E−u

[(∫ τ

0e−γ t |ut |dt +

∫ τ

0e−γ tdit

)2]+ E

[∣∣M−uτ

∣∣3 + [M−u

τ

]−3]< ∞. (7.12)

We now characterize the optimal control u. Denote by WA,ut and W

A,ut the

agent’s remaining utility and the discounted remaining utility, respectively:

WA,ut = E−u

t

[∫ τ

t

e−γ (s−t)[λusvds + dis] + e−γ (τ−t)R

], W

A,ut = e−γ tW

A,ut .

(7.13)

By (7.12) and the Martingale Representation Theorem of Lemma 10.4.6, there existsZA,u such that

WA,ut = e−γ τR +

∫ τ

t

e−γ s[λusvds + dis] −∫ τ

t

e−γ svZA,us dB−u

s .

This leads to

dWA,ut = e−γ tutv

[Z

A,ut − λ

]dt − e−γ tdit + e−γ t vZ

A,ut dBt and

WA,uτ = e−γ τR.

(7.14)

Equivalently,

dWA,ut = γW

A,ut dt + utv

[Z

A,ut − λ

]dt − dit + vZ

A,ut dBt and WA,u

τ = R.

(7.15)

7.2 Agent’s Problem 119

Theorem 7.2.3 Assume Assumptions 7.1.1 and 7.2.2 hold. For any (τ, i) ∈ A, u ∈A(τ, i) is optimal if and only if

ZA,ut = λ when ut > 0 and Z

A,ut ≥ λ when ut = 0. (7.16)

Proof (i) We first prove the sufficiency. Assume u ∈A(τ, i) satisfies (7.16). For anyu ∈ A(τ, i). Denote

�u := u − u, �WAt := W

A,ut − W

A,ut , �ZA

t := ZA,ut − Z

A,ut .

Then,

�WA0− = �WA

0 =∫ τ

0e−γ t v

[λ − Z

A,ut

]�utdt −

∫ τ

0e−γ t v�ZA

t dB−ut . (7.17)

By (7.16) we have [λ − Z

A,ut

]�ut ≥ 0,

and thus

�WA0− ≥ −

∫ τ

0e−γ t v�ZA

t dB−ut .

Note that

E−u

[(∫ τ

0

∣∣e−γ tZA,ut

∣∣2dt

) 12]

= E−u

[(M−u

τ

)−1M−u

(∫ τ

0

∣∣e−γ tZA,ut

∣∣2dt

) 12]

≤ (E−u

[(M−u

τ

)−2(M−u

)2]) 12

(E−u

[∫ τ

0

∣∣e−γ tZA,ut

∣∣2dt

]) 12

= (E

[(M−u

τ

)−1(M−u

)2]) 12

(E−u

[∫ τ

0

∣∣e−γ tZA,ut

∣∣2dt

]) 12

≤ (E

[(M−u

τ

)−3]) 16(E

[(M−u

τ

)3]) 13

(E−u

[∫ τ

0

∣∣e−γ tZA,ut

∣∣2dt

]) 12

< ∞, (7.18)

thanks to (7.12). Then,

E−u

[∫ τ

0e−γ t v�ZA

t dB−ut

]= 0. (7.19)

Therefore, �WA0− ≥ 0 for any u ∈A(τ, i), that is, u is optimal.

(ii) We next prove necessity. Assume u ∈A(τ, i) is optimal. Set

θt := (ut ∧ 1)sgn(λ − Z

A,ut

) + 1{ut=0}1{λ>ZA,ut }.

Then,


ut + θt ≥ 0.

For each n, denote

τn := inf

{t ≥ 0 : M−(u+θ)

t ≥ nM−ut or M

−(u+θ)t ≤ 1

nM−u

t

}∧ τ.

Then, τn ↑ τ . Set

unt = ut + θt1[0,τn](t).

One can easily check that un satisfies Assumption 7.2.2 and thus un ∈ A(τ, i).By (7.17) we have

WA,u0− − W

A,un

0− = −∫ τn

0e−γ t v

[λ − Z

A,ut

]θtdt −

∫ τ

0e−γ t v�ZA

t dB−ut

= −∫ τn

0e−γ t v

[(ut ∧ 1)

∣∣λ − ZA,ut

∣∣ + 1{ut=0}(λ − Z

A,ut

)+]dt

−∫ τ

0e−γ t v�ZA

t dB−ut .

Since u is optimal, then (7.19) implies that

0 ≤ WA,u0− − W

A,un

0−

= −E−u

[∫ τn

0e−γ t v

[(ut ∧ 1)

∣∣λ − ZA,ut

∣∣ + 1{ut=0}(λ − Z

A,ut

)+]dt

].

This proves (7.16) on [0, τn]. Sending n → ∞, we obtain the result. �

Remark 7.2.4 If we assume u ≤ C0 for all u ∈ A(τ, i), then one has �u ≥ 0 whenu = C0. Following similar arguments, one can easily see that u ∈A(τ, i) is optimalif and only if

ZA,ut = λ when 0 < ut < C0;

ZA,ut ≥ λ when ut = 0; and

ZA,ut ≤ λ when ut = C0.

(7.20)

The next result shows that truth-telling is always one possible optimal strategyfor the agent.

Corollary 7.2.5 Assume Assumptions 7.1.1 and 7.2.2 hold. A contract (τ, i) ∈A isimplementable if and only if ZA,0 ≥ λ. In particular, for any implementable (τ, i) ∈A, u = 0 is optimal.

Proof If ZA,0 ≥ λ, by Theorem 7.2.3 we know that u = 0 is optimal, and in particu-lar, (τ, i) is implementable. On the other hand, assume (τ, i) ∈ A is implementableand u ∈A(τ, i) is an arbitrary optimal control. Then, (7.16) holds. This implies that

ut

[Z

A,ut − λ

] = 0,

7.3 Principal’s Problem 121

and thus, by (7.14),

WA,ut = e−γ τR +

∫ τ

t

e−γ sdis −∫ τ

t

e−γ svZA,us dBs.

This is the same BSDE as for WA,0. By (7.18) with u = 0, we see that WA,u0− =

WA,00− . That is, u = 0 is also optimal. �

Considering now WA,u instead of WA,u, we state

Corollary 7.2.6 Assume Assumptions 7.1.1 and 7.2.2 hold. Suppose that

ZA ≥ λ, E

[∫ τ

0

∣∣e−γ tZAt

∣∣2dt

]< ∞, WA

τ = R,

where

dWAt = γWA

t dt − dit + vZAt dBt , WA

0− = wA. (7.21)

Then, u ∈ A(τ, i) is optimal if and only if ut = 0 when ZAt > λ, and in this case we

have WA,u0− = WA

0− = wA. In particular, if ZA ≡ λ, then any u ∈ A(τ, i) is optimal.That is, the agent is indifferent with respect to the choice of u.

7.3 Principal’s Problem

We now investigate the principal’s problem (7.9). The principal’s utility (7.8) can bewritten as

WP,τ,i,u0− := W

P,τ,i,u0 − �i0

:= E−u

[∫ τ

0e−rs

[(μ − utv)dt − dit

] + e−rτL

]− �i0. (7.22)

7.3.1 Principal’s Problem Under Participation Constraint

Fix the agent’s optimal utility at time zero at value wA. We modify the principal’sproblem by assuming that the agent’s promised utility has to be larger than R at alltimes (otherwise, he would quit and take his “outside option”):

b(wA) := sup(τ,i)∈A(wA)

supu∈A0(τ,i)

WP,τ,i,u0− (7.23)

where, for wA ≥ R,

A(wA) := {(τ, i) ∈ A : WA

0− = wA and WAt ≥ R,0 ≤ t ≤ τ

};A0(τ, i) := the set of the agent’s optimal controls u ∈A(τ, i).

(7.24)


We need the latter notation because the agent’s optimal u may not be unique. Inparticular, by Corollary 7.2.5, u ≡ 0 ∈ A0(τ, i) for any (τ, i) ∈ A(wA). We notethat, by Corollary 7.2.6, process WA depends only on (τ, i)—it stays the same fordifferent u ∈A0(τ, i).

We first have:

Lemma 7.3.1 For any wA ≥ R, A(wA) is not empty.

Proof Fix any R ≥ wA. Let ZAt := λ and i be the reflection process which keeps

WAt within [R, R], and the contract terminates once WA

t hits R. That is, i is thesmallest increasing process such that

WAt = wA +

∫ t

0γWA

s ds − it + λvBt

stays within [R, R], and

τ := inf{t : WA

t = R}.

We will show that (τ, i) ∈A(wA). We first show that τ < ∞, a.s. Denote p(wA) :=P(τ = ∞|WA

0 = wA). Clearly, p is increasing in wA. Then,

p(R) = P(WA

t > R for all t ≥ 0|WA0 = R

) = E[1{inf0≤t≤1 WA

t >R}p(WA

1

)|WA0 = R

]≤ E

[1{inf0≤t≤1 WA

t >R}p(R)|WA0 = R

] = P(

inf0≤t≤1

WAt > R

)p(R).

It is obvious that P(inf0≤t≤1 WAt > R) < 1. Then, p(R) = 0 and thus p(wA) = 0.

We next show that

E

[(∫ τ

0e−γ tdit

)2]< ∞.

In fact,

e−γ tWAt = wA −

∫ t

0e−γ sdis + λv

∫ t

0e−γ sdBs.

Then, ∫ τ

0e−γ sdis = wA − e−γ τR + λv

∫ τ

0e−γ sdBs,

and thus,

E

[(∫ τ

0e−γ tdit

)2]= E

[∣∣wA − e−γ τR∣∣2 + |λv|2

∫ τ

0e−2γ sds

]

≤ |wA − R|2 + |λv|22γ

< ∞. (7.25)

Finally, by definition WAt ≥ R for t ≤ τ . Therefore, (τ, i) ∈ A(wA). �


Lemma 7.3.2 For any wA ≥ R and (τ, i) ∈ A(wA), it must hold that

τ = inf{t ≥ 0 : WA

t ≤ R}. (7.26)

Consequently,

b(R) = L. (7.27)

Proof By Corollary 7.2.5, for ZA such that (7.21) holds, we have ZA ≥ λ andWA

τ = R. Denote

τ := inf{t ≥ 0 : WA

t ≤ R}.

Then, τ ≤ τ . Since WA ≥ R and WA is right continuous, we have WAτ

= R. Notethat

WAτ+δ

− WAτ

=∫ τ+δ

τ

γWAs ds −

∫ τ+δ

τ

dis +∫ τ+δ

τ

vZAs dBs

≤∫ τ+δ

τ

vZAs

[dBs − γWA

s /vZAs ds

],

and that γWAs /vZA

s is locally bounded. Then, by applying Girsanov’s theorem wesee that WA can not satisfy the IR constraint WA

t ≥ R for t > τ , and therefore,τ = τ . �

By the above results, we see that (τ, i) is in one-to-one correspondence with(ZA, i). Let A(wA) denote the set of FB -adapted processes (ZA, i) such that

(i) i is non-decreasing and right continuous with left limits;(ii) ZA ≥ λ and

∫ t

0 |ZAs |2ds < ∞, P -a.s. for any t ;

(iii) for WA and τ defined by (7.21) and (7.26), it holds that τ < ∞, a.s. andWA

τ = R;(iv) E[∫ τ

0 |e−γ tZAt |2dt] < ∞.

Moreover, for each (ZA, i) ∈ A(wA), let A0(ZA, i) denote the set of u ∈A(τ, i) for

the corresponding τ such that ut = 0 whenever ZAt > λ. Then, it is clear that, for

wA ≥ R,

b(wA) = sup(ZA,i)∈A(wA)

supu∈A0(Z

A,i)

E−u

[−�i0 +

∫ τ

0e−rs

[(μ − utv)dt − dit

]

+ e−rτL

]. (7.28)

Lemma 7.3.3 Let ZA, i, τ satisfy E[∫ τ

0 |e−γ t ZAt |2dt + | ∫ τ

0 e−γ tdit |2] < ∞. Forany wA ≥ R, denote

dWAt = γ WA

t dt − dit + vZAt dBt , WA

0− = wA.

If WAt ≥ R, 0 ≤ t ≤ τ , then there exists (ZA, i) ∈ A(wA) such that ZA

t = ZAt , it =

it , 0 ≤ t ≤ τ .


Proof On [0, τ ] set ZAt := ZA

t , it := it , and thus WAt = WA

t . For t > τ , let ZAt := λ

and i be the reflection process (which is continuous) such that the following processWA

t stays within [R,WAτ

]:dWA

t = γWAt dt − dit + λvdBt , WA

τ = WAτ .

By Lemma 7.3.1, we know τ := inf{t : WAt = R} < ∞ a.s. Moreover, by (7.25) we

see that

Eτ

[(∫ τ

τ

e−γ (s−τ )dis

)2]≤ ∣∣WA

τ − R∣∣2 + |λv|2

2γ≤ ∣∣WA

τ

∣∣2 + |λv|22γ

.

By our conditions it is clear that

E[∣∣e−γ τ WA

τ

∣∣2]< ∞.

Then,

E

[(∫ τ

0e−γ sdis

)2]

= E

[(∫ τ

0e−γ sdis + e−γ τ

∫ τ

τ

e−γ (s−τ )dis

)2]

≤ 2E

[(∫ τ

0e−γ sdis

)2

+ e−2γ τ

[∣∣WAτ

∣∣2 + |λv|22γ

]]< ∞.

Then, it follows that (ZA, i) ∈ A(wA). �

Next, following standard arguments, one has the following Dynamic Program-ming Principle for the problem (7.28), for any stopping time τ :

b(wA) = sup(ZA,i)∈A(wA)

supu∈A0(Z

A,i)

E−u[Gτ∧τ ] (7.29)

where

Gt := −�i0 +∫ t

0e−rs

[(μ − usv)ds − dis

] + e−rt b(WA

t

). (7.30)

If b is sufficiently smooth, by Itô’s formula we have

ertdGt =[μ − utv

[1 + ZA

t b′(WAt

)] + γWAt b′(WA

t

) + 1

2

(vZA

t

)2b′′(WA

t

)

− rb(WA

t

)]dt − [

1 + b′(WAt−

)]dit + vZA

t b′(WAt

)dB−u

t

=[μ − utv

[1 + λb′(WA

t

)] + γWAt b′(WA

t

) + 1

2

(vZA

t

)2b′′(WA

t

)

− rb(WA

t

)]dt − [

1 + b′(WAt−

)]dit + vZA

t b′(WAt

)dB−u

t , (7.31)


thanks to the fact that ut [ZAt − λ] = 0. Thus, noting that WA

0 = wA − �i0, we get

b(wA) = sup(ZA,i)∈A

supu∈A0(Z

A,i)

E−u

[b(wA − �i0) − �i0

−∫ τ∧τ

0e−rt

[1 + b′(WA

t−)]

dit

+∫ τ∧τ

0e−rt

[μ − utv

[1 + λb′(WA

t

)] + γWAt b′(WA

t

)

+ 1

2

(vZA

t

)2b′′(WA

t

) − rb(WA

t

)]dt

]. (7.32)

7.3.2 Properties of the Principal’s Value Function

In this subsection we derive heuristically the form of the principal’s value function b.First, the principal can always pay a lump-sum of di > 0 to the agent, which meansthat we have b(w) ≥ b(w − di) − di. This would imply b′(w) ≥ −1. Moreover,as long as we have strict inequality, there will be no payments. More precisely,for any wA ≥ R and any w > wA, applying (7.29) to b(w) by setting τ := 0 and�i0 := w − wA, we get

b(w) ≥ −�i0 + b(WA

0

) = b(w − �i0) − �i0 = b(wA) − w + wA. (7.33)

Sending w ↓ wA, this implies that, assuming b′ exists,

b′(wA) ≥ −1 for all wA ≥ R. (7.34)

Next, we guess that

b ∈ C2 and is concave. (7.35)

Denote

R∗ := inf{w ≥ R : b′(w) ≤ −1

}. (7.36)

Since b′ is decreasing, then b′(w) ≤ −1 for all w ≥ R∗. This, together with (7.34),implies that

on[R∗,∞)

, b′(wA) = −1 and thus b(wA) = b(R∗) − (

wA − R∗). (7.37)

We now consider w ∈ [R,R∗). Setting

ZA := λ, i := 0, u := 0

and plugging τ := δ > 0 into (7.32), we have

E

[∫ τ∧δ

0e−rt

[μ + γWA

t b′(WAt

) + 1

2(vλ)2b′′(WA

t

) − rb(WA

t

)]dt

]≤ 0.


Dividing by δ and sending δ → 0, we get

μ + γwAb′(wA) + 1

2v2λ2b′′(wA) − rb(wA) ≤ 0. (7.38)

On the other hand, for any (ZA, i, u), plugging τ := δ into (7.32), we get

sup(ZA,i)∈A,u∈A0(Z

A,i)

E−u

[−

∫ τ∧δ

0e−rt

[1 + b′(WA

t

)]dit

+∫ τ∧δ

0e−rt

[μ − utv

[1 + λb′(WA

t

)] + γWAt b′(WA

t

)

+ 1

2

(vZA

t

)2b′′(WA

t

) − rb(WA

t

)]dt

]= 0.

By (7.34), λ ≤ 1, and b′′ ≤ 0, we have

sup(ZA,i)∈A,u∈A0(Z

A,i)

E

[M−u

τ∧δ

∫ τ∧δ

0e−rt

[μ + γWA

t b′(WAt

)

+ 1

2(vλ)2b′′(WA

t

) − rb(WA

t

)]dt

]≥ 0.

Dividing by δ and sending δ → 0, formally we obtain

μ + γwAb′(wA) + 1

2(vλ)2b′′(wA) − rb(wA) ≥ 0.

This, together with (7.38), leads to

μ + γwb′(w) + 1

2λ2v2b′′(w) − rb(w) = 0, w ∈ [R,R∗). (7.39)

Finally, by (7.37) we have b′(R∗) = −1 and b′′(R∗) = 0. The condition b′(R∗) =−1 is the “smooth-pasting” or “smooth-fit” condition, guaranteeing that the deriva-tive from the left and the right agree at w = R∗. Moreover, in order to have smoothfit for the second derivative, b′′(R∗) = 0, the differential equation for b implies alsothe condition

rb(R∗) + γR∗ = μ. (7.40)

Intuitively, the payments are postponed until the expected return of the project μ iscompletely used up by the principal’s and the agent’s expected returns.

In conclusion, given (7.35), function b should be determined by (7.37) and (7.39),together with the boundary condition (7.27) and the free boundary conditions (7.40).We state the precise result next.

7.3.3 Optimal Contract

The main result of this chapter is:


Theorem 7.3.4 Assume Assumptions 7.1.1 and 7.2.2 hold. Consider the ODE sys-tem (7.37), (7.39), and (7.40):

μ + γwb′(w) + 1

2λ2v2b′′(w) − rb(w) = 0, w ∈ [R,R∗);

on[R∗,∞)

, b′(wA) = −1 and thus b(wA) = b(R∗) − (

wA − R∗);b(R) = L, rb

(R∗) + γR∗ = μ;

and assume it has a concave solution b ∈ C2. Then,

(i) b is the value function defined by (7.23) (or equivalently by (7.28)).(ii) When wA ∈ [R,R∗], truth-telling is optimal, u ≡ 0. Moreover, it is optimal to

set

ZAt ≡ λ,

and the payments i to be the reflection process which keeps WAt within [R,R∗].

The optimal contract terminates once WAt hits R. That is, i is the smallest

increasing process such that

WAt = wA +

∫ t

0rWA

s ds − it + λBt

stays within [R,R∗]. In particular, when WAt ∈ (R,R∗), dit = 0.

(iii) When wA > R∗, then the optimal contract pays an immediate payment ofwA − R∗ to the agent, and the contract continues with the agent’s new ini-tial utility R∗.

Proof Let b denote the solution to the ODE system and b denote the value functiondefined by (7.28). We first show that

b ≤ b. (7.41)

To see that, given (ZA, i) ∈ A(wA) and u ∈ A0(τ, i), recall the process G in (7.29)and its dynamics (7.31). By (7.37) and the assumption that b is concave, we see thatb′(wA) ≥ −1 for wA ∈ [R,R∗] and thus

1 + λb′(WAt

) ≥ 1 + b′(WAt

) ≥ 0.

This, together with the assumptions that b is concave and ZA ≥ λ, implies that

ertdGt ≤[μ + γWA

t b′(WAt

) + 1

2λ2v2b′′(WA

t

) − rb(WA

t

)]dt

+ ZAt b′(WA

t

)dB−u

t .

When WAt ∈ [R,R∗], by (7.39) we have

μ + γWAt b′(WA

t

) + 1

2λ2v2b′′(WA

t

) − rb(WA

t

) = 0.

When WAt > R∗, by (7.37), (7.40), and the assumption that γ > r , we have


μ + γWAt b′(WA

t

) + 1

2λ2v2b′′(WA

t

) − rb(WA

t

)= rb

(R∗) + γR∗ − γWA

t − r[b(R∗) − WA

t + R∗] = [r − γ ][WAt − R∗] < 0.

Therefore, in all the cases we have

ertdGt ≤ ZAt b′(WA

t

)dB−u

t .

Since b′ is bounded, G is a P −u-super-martingale. Notice that b(wA) = G0 andb(WA

τ ) = b(R) = L. We have then

b(wA) = G0 ≥ E−u[Gτ ]. (7.42)

This, together with (7.29), setting τ := τ , proves (7.41).We next prove b(wA) = b(wA) for wA ∈ [R,R∗]. Let (ZA, i, u) be specified as

in (ii). By Lemma 7.3.1 we see that (ZA, i) ∈ A(wA).Since dit > 0 implies that WA

t = R∗ and thus b′(WAt ) = −1, we have (1 +

b′(WAt ))dit = 0. Moreover, since WA

t ∈ [R,R∗], by (7.39) and (7.31) we see that

ertdGt = ZAt b′(WA

t

)dBt .

This, together with the fact that b′ is bounded, implies that G is a P -martingale.Then, we have

b(wA) = G0 = E[Gτ ].Thus, b(wA) = b(wA) for wA ∈ [R,R∗] and (ZA, i, u) given in (ii) is an optimalcontract.

Finally, for wA > R∗, note that the value function should satisfy (7.33), and thenby (7.40) we have b(wA) ≥ b(wA). This, together with (7.41), implies that b(wA) =b(wA). It is obvious that the contract described in (iii) is optimal. �

Remark 7.3.5 In this remark we discuss how to find the above function b. Considerthe following elliptic ODE with parameter θ ≥ −1:⎧⎨

⎩μ + γwb′

θ (w) + 1

2λ2v2b

′′θ (w) − rbθ (w) = 0, w ∈ [R,∞);

bθ (R) = L, b′θ (R) = θ.

(7.43)

By standard results in the ODE literature, the above ODE has a unique smoothsolution bθ . By Assumption 7.1.1(i),

1

2λ2v2b

′′θ (R) = −[

μ + γRb′θ (R) − rbθ (R)

]= −[μ + γRθ − rL] ≤ γR + rL − μ ≤ 0.

Denote

R∗θ := inf

{w ≥ R : b′′

θ (w) = 0}.

If we can find θ ≥ −1 such that

R∗θ < ∞ and b′

θ

(R∗

θ

) = −1, (7.44)

7.4 Implementation Using Standard Securities 129

then one can check straightforwardly that

b(w) := bθ (w)1[R,R∗θ ](w) + [

bθ

(R∗

θ

) + R∗θ − w

]1(R∗

θ ,∞) (7.45)

satisfies all our requirements.

7.4 Implementation Using Standard Securities

We now want to show that the above contract can be implemented using real-worldsecurities, namely, equity, long-term debt and credit line. The implementation willbe accomplished using the following:

– The firm starts with initial capital K and possibly an additional amount neededfor initial dividends or cash reserves.

– The firm has access to a credit line up to a limit of CL. The interest rate onthe credit line balance is rC . The agent decides on borrowing money from thecredit line and on repayments to the credit line. If the limit CL is reached, thefirm/project is terminated.

– Shareholders receive dividends which are paid from cash reserves or the creditline, at the discretion of the agent.

– The firm issues a long (infinite) term debt with continuous coupons paying at ratex, with face value of the debt equal to D = x/r . If the firm cannot pay a couponpayment, the project is terminated.

The agent will be paid by a fraction of dividends. We assume that once the projectis terminated the agent does not receive anything from his holdings of equity. Hereis the result that shows precisely how the optimal contract is implemented.

Theorem 7.4.1 Suppose that the credit line has interest rate rC = γ , and that thelong-term debt satisfies

x = rD = μ − γR/λ − γCL. (7.46)

Assume the dividends δtdt are paid only at the times the credit line balance hitszero, so that λδt is the reflection process that keeps the credit line above zero. If theagent is paid by a proportion λδtdt of the firm’s dividends, he will not misreportthe cash flows, and will use them to pay the debt coupons and the credit line beforeissuing dividends. Denoting the current balance of the credit line by Mt , the agent’sexpected utility process satisfies

WAt = R + λ

(CL − Mt

). (7.47)

If in addition

CL = (R∗ − R

)/λ (7.48)

then the above capital structure of the firm implements the optimal contract.


Proof Denote by δt the cumulative dividends process. By that, we mean that δtdt

is equal to whatever money is left after paying the interest γMtdt on the credit lineand the debt coupons xdt . Since the total amount of available funds is equal to thebalance of the credit line M plus the reported profit X, and since M + X is dividedbetween the credit line interest payments, debt coupon payments and dividends, wehave

dMt = γMtdt + xdt + dδt − dXt .

With WAt as in (7.47), and from x = rD and (7.46), we have

dWAt = −λdMt = γWA

t dt − λdδt + λdBt .

If we set dit = λdδt , then this corresponds to the agent’s utility with zero savingsand ZA ≡ λ, which implies that the agent will not have incentives to misreport.Moreover, since the dividends are paid when Mt = 0, which, by (7.48) is equiva-lent to WA

t = R∗, we see that the optimal strategy is implemented by this capitalstructure. �

Remark 7.4.2 (i) Truth-telling is a consequence of providing the agent with a frac-tion λ of dividends, and giving him control over the timing of dividends. The agentwill not pay the dividends too soon because of (7.47)—if he did empty the creditline instantaneously at time t to pay the dividends and then immediately default, hewould be getting WA

t in expected utility, which is also what he is getting by waitinguntil the credit line balance is zero. He does pay the dividends once that happensbecause γ > r .

(ii) The choice of credit line limit CL resolves the trade-off between delayingthe agent’s payments and delaying termination. The level of debt payments x = rD

cannot be too high in order to ensure that the agent does not use the credit line toosoon; it cannot be too low either, otherwise the agent would save excess cash evenafter the credit line is paid off, in order to delay termination.

(iii) It is possible to have D < 0, which is to be interpreted as a margin accountthe firm may have to keep in order to have access to the credit line. This accountearns interest r , cannot be withdrawn from, and is exhausted by creditors in case oftermination.

7.5 Comparative Statics

The above framework enables us to get many economics conclusions, either by ana-lytic derivations, or by numerical computations. We list here some of those reportedin DeMarzo and Sannikov (2006).

7.5 Comparative Statics 131

7.5.1 Example: Agent Owns the Firm

Suppose λ = 1. In this case the agent gets all the dividends, and the firm is financedby debt. It can be verified that D = b(R∗). Suppose also that WA

0 is chosen so thatb(WA

0 ) = K , the lowest payoff the investors require. With extremely high volatilityv, it may happen that such WA

0 does not exist, and no contract is offered. In thecases this is not a problem, numerical examples show that with higher volatility theprincipal’s expected utility b(w) gets smaller, the credit line limit CL gets larger, thedebt level D = b(R∗) gets smaller. In fact, D becomes negative for very high levelsof volatility. The required capital K and the margin balance (−D) are financed bya large initial draw of R∗ − WA

0 on the credit line (recall that dWA = −λdM). Themargin balance pays interest to the project, and this provides incentives for the agentto keep running the firm. This interest is received even after the credit line is paidoff, and thus, the upfront financing by investors of the margin balance is a way toguarantee a long-term commitment by the investors to the firm.

With medium volatility we may have 0 < D < K , so that part of the initial capitalK is raised from debt, and part by a draw R∗ − WA

0 on the credit line. With verylow volatility, we may have WA

0 > R∗, so that an immediate dividend of WA0 − R∗

is paid.

7.5.2 Computing Parameter Sensitivities

We now show how to compute partial derivatives of the values determining theoptimal contract with respect to a given parameter θ . We first have the followingFeynman–Kac type result:

Lemma 7.5.1 Suppose the process W has the dynamics

dWt = γWtdt + λvdBt − dit

where i is a local-time process that makes W reflect at R∗. The process W is stoppedat time τ = min{t : Wt = R}. Given a function g bounded on interval [R,R∗] andconstants r, k,L, suppose a function G defined on [R,R∗] solves the ordinary dif-ferential equation

rG(w) = g(w) + γwG′(w) + 1

2λ2v2G′′(w) (7.49)

with boundary conditions

G(R) = L, G′(R∗) = −k.

Then, G can be written as

G(w) = EW0=w

[∫ τ

0e−rt g(Wt )dt − k

∫ τ

0e−rt dit + e−rτL

]. (7.50)


Proof Define the process

Ht =∫ t

0e−rsg(Ws)ds − k

∫ t

0e−rsdis + e−rtG(Wt).

Using Itô’s rule, we get

ertdHt =[g(Wt) + γWtG

′(Wt) + 1

2λ2v2G′′(Wt) − rG(Wt)

]dt

− (k + G′(Wt)

)dit + G′(Wt )λvdBt .

Since G satisfies the ODE, the dt term is zero. So is the dit term, because G′(R∗) =−k and the process i changes only when Wt = R∗. Moreover, since also G′(w) isbounded on [R,R∗], the process Ht is a martingale, and E[Ht ] = H0 = G(W0).Moreover, as G is bounded on [R,R∗], then also E[Hτ ] = G(W0). �

Let θ be one of the parameters L,μ,γ, v2 or λ, and denote by bθ (w) the functionrepresenting the optimal principal’s utility given that parameter. We can computethen its derivative using

Proposition 7.5.2 We have

∂θbθ (w)

= EWA0 =w

[∫ τ

0e−rt

(∂θμ + (∂θγ )WA

t b′θ

(WA

t

) + 1

2

[∂θ

(λ2v2)]b′′

θ

(WA

t

))dt

+ e−rτ ∂θL

].

Proof Denote by b(w) = bθ,R∗(w) the principal’s expected utility function given θ

and the reflecting point R∗. This function satisfies

rb(w) = μ + γwb′(w) + 1

2λ2v2b′′(w) (7.51)

with boundary conditions b(R) = L, b′(R∗) = −1. Denote by R∗(θ) the value ofR∗ which maximizes bθ,R∗(WA

0 ), so that bθ = bθ,R∗(θ). By the Envelope Theoremwe have

∂θbθ (w) = ∂θbθ,R∗(θ)(w) = ∂θbθ,R∗(w)∣∣R∗=R∗(θ)

.

Using this and differentiating (7.51) with respect to θ at R∗ = R∗(θ), we get

r∂θbθ (w) = ∂θμ + (∂θγ )wb′θ (w) + γw∂w∂θbθ (w) + 1

2

[∂θ

(λ2v2)]b′′

θ (w)

+ 1

2λ2v2∂2

w2∂θbθ (w) (7.52)

with boundary conditions ∂θbθ (R) = ∂θL, ∂w∂θbθ (R∗) = 0. The statement follows

from Lemma 7.5.1. �

7.5 Comparative Statics 133

Using the knowledge of ∂θb(w), we can find the effect on θ on debt and creditline by differentiating the boundary condition rb(R∗) + γR∗ = μ, as well as thedefinition of the agent’s starting value, for example b(WA

0 ) = K .As an example, we get that

∂Lb(w) = EW0=w

[e−rτ

].

Moreover, differentiating the boundary condition (considering b as a function of twovariables, L and w), we get

r[∂Lb

(R∗) + b′(R∗)∂LR∗] + γ ∂LR∗ = 0.

Since b′(R∗) = −1, we get

∂LR∗ = − r

γ − rEWA

0 =R∗[e−rτ

].

Thus, larger L means shorter time until paying dividends, and hence shorter creditline, because liquidation is less inefficient. One can similarly compute ∂θb(w) forother θ , and also the sensitivities of debt and credit line to θ , and we report some ofthe conclusions below. Since θ = R was not included in the cases above, let us alsomention that we have

∂Rb(w) = −b′(R)EWA0 =w

[e−rτ

].

This is because when changing the agent’s liquidation value R by dR while simul-taneously changing the principal’s liquidation value L by b′(R)dR, the principal’sexpected utility will not change.

7.5.3 Some Comparative Statics

Using the above method one can compute many comparative statics, reported inDeMarzo and Sannikov (2006). We mention some conclusions next:

– As L increases, CL decreases, since termination is less undesirable.– As R increases, CL and debt payments decrease to reduce the agent’s desire to

default sooner.– As μ increases, CL increases to delay termination, and debt increases since the

cash flows are higher.– As the agent’s discount rate γ increases, CL decreases as the agent is more im-

patient to start consuming. The debt value can either increase or decrease.– As volatility v increases, CL increases and debt decreases, as discussed in the

example above.– If we choose the highest possible WA

0 , that is, such that b(WA0 ) = K , WA

0 in-creases with L and μ and decreases with v2, γ, λ and R. Similarly, if we choosethe highest possible amount for b(WA

0 ), it behaves in the same way.


7.6 Further Reading

This chapter is based on DeMarzo and Sannikov (2006). Biais et al. (2007) obtainequivalent results in the limit of a discrete model. Earlier discrete-time dynamicagency models of the firm include Spear and Srivastava (1987), Leland (1998),Quadrini (2004), DeMarzo and Fishman (2007a, 2007b). Survey paper Sannikov(2012) provides additional references to recent papers.

References

Biais, B., Mariotti, T., Plantin, G., Rochet, J.-C.: Dynamic security design: convergence to contin-uous time and asset pricing implications. Rev. Econ. Stud. 74, 345–390 (2007)

DeMarzo, P.M., Fishman, M.: Optimal long-term financial contracting. Rev. Financ. Stud. 20,2079–2128 (2007a)

DeMarzo, P.M., Fishman, M.: Agency and optimal investment dynamics. Rev. Financ. Stud. 20,151–188 (2007b)

DeMarzo, P.M., Sannikov, Y.: Optimal security design and dynamic capital structure in acontinuous-time agency model. J. Finance 61, 2681–2724 (2006)

Leland, H.E.: Agency costs, risk management, and capital structure. J. Finance 53, 1213–1243(1998)

Quadrini, V.: Investment and liquidation in renegotiation-proof contracts with moral hazard.J. Monet. Econ. 51, 713–751 (2004)

Sannikov, Y.: Contracts: the theory of dynamic principal-agent relationships and the continuous-time approach. Working paper, Princeton University (2012)

Spear, S., Srivastrava, S.: On repeated moral hazard with discounting. Rev. Econ. Stud. 53, 599–617 (1987)

[springer finance] contract theory in continuous-time models || an application to capital structure...

Documents