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Corporate Finance:Asymmetric information and capital structure – signaling
Yossi Spiegel
Recanati School of Business
Ross, BJE 1977
“The Determination of Financial Structure: The Incentive-Signalling
Approach”
Corporate Finance 3
The model� The timing:
� Cash flow is x ~ U[0,T], where T ∈ {L,H}, L < H
� T is private info. to the firm
� The capital market believes that T = H with prob. γ
� In case of bankruptcy, the manager bears a personal loss C
� The manager’s utility: U = V – C x Prob. of bankruptcy
The firm issues debt with face value D
Stage 0 Stage 2Stage 1
The capital market observes D and updates the belief about the firm’s type
Cash flow is realized
Corporate Finance 4
The full information case� The value of the firm when T is common knowledge:
� With uniform dist.:
� The manager’s utility:
� The manager will not issue debt
T
DCxDFCxU TT −=×−= ˆ)(ˆ
( ) T
T
D
T
D
D
T xxdFDxxDdFxxdFV ˆ)()()(0
=−+
+= ∫∫∫
2/ˆ TxT =
Corporate Finance 5
Asymmetric information
� DT* is the debt level of type T
� B* is the prob. that the capital market assigns to the firm being of type T
� Perfect Bayesian Equilibrium (PBE), (DH*,DL*,B*):� DH* and DL* are optimal given B*
� B* is consistent with the Bayes rule
Corporate Finance 6
Bayes rule
� In our case, two levels of D are chosen. Suppose the probability that each type plays them is as follows:
� Having observed D1 the capital market believes that the firm is type H with prob.:
� In a sep. equil., h1 = 1 and l1 = 0. In a pooling equil., h1 = l1 = 1
( ) ( )( )BP
APABPBAP
)(|| =
l2l1Type L (1−γ)
h2h1Type H (γ)
D2D1
( ) ( )( ) )1(
)(||
11
1
1
11 γγ
γ−+
==lh
h
DP
HPHDPDHP
Corporate Finance 7
Separating equilibria: DH* ≠ DL*
� The belief function:
� In a separating equil., DL* = 0 because type L cannot boost V by issuing debt
� DL* = 0 ⇒ B*(0) = 0 ⇒ V(0) = L/2
=
=
=
wo
DD
DD
B L
H
/'
*0
*1
*
γ
Corporate Finance 8
Separating equilibrium� The IC of H:
� The IC of L:
� Indifference of type T between V and D:
{
0 Dwhen H of Payoff
D issuesit when H of Payoff
2
=
≥×−L
H
DCV43421
{ 43421
D issuesit when L of Payoff
0 Dwhen L of Payoff
2 L
DCV
L×−≥
=
{ T
DC
LV
T
DCV
L×+=⇒×−=
=
22
V valueinducesit ifD issuing when Payoff
0D ifPayoff
43421
Corporate Finance 9
The set of separating equilibria
D
L
DC
LV +=
2 H
DC
LV +=
2
2
HV =
2
LV =
V
D* D**
sep. equil.
D* is defined byL/2+CxD/L = H/2⇒ D* = (H-L)L/2C
Corporate Finance 10
The DOM criterion� D ≥ D* is a dominated strategy for type L: D = 0 always guarantees L a higher
payoff no matter what the capital market believes
� The DOM criterion:
� The idea: type L will never play D ≥ D* while type H might. Hence, if D ≥ D*, then the firm’s type must be H
� The DOM criterion still does not determine the beliefs for D < D* and D ≠ 0
� Under the DOM criterion: DL* = 0, DH*=D*
≥
=
=
=
wo
DD
DD
DD
BL
H
/'
*1
*0
*1
*
γ
Corporate Finance 11
Separating equilibria under the DOM criterion
D
L
DC
LV +=
2 H
DC
LV +=
2
2
HV =
2
LV =
V
D* D**
DL* = 0, DH*=D*
Corporate Finance 12
Pooling equilibria: DH* = DL* = Dp*
� The belief function:
� In a pooling equil., the choice of Dp* is uninformative; hence
=
=wo
DDB
p
/'
**
γγ
( )2
12
ˆ LHV ×−+×= γγ
Corporate Finance 13
The set of pooling equilibria
D
L
DC
LV +=
2 H
DC
LV +=
2
2
HV =
2
LV =
V
D* D**
poolingequil.
VV ˆ=
Corporate Finance 14
The DOM criterion
� The DOM criterion:
� The DOM criterion eliminates some pooling equilibria but not all
≥
=
=
wo
DD
DD
B
p
/'
*1
*
*
γ
γ
Corporate Finance 15
The set of pooling equilibria
D
L
DC
LV +=
2 H
DC
LV +=
2
2
HV =
2
LV =
V
D* D**
poolingequil.
VV ˆ=
Corporate Finance 16
The intuitive criterion� Due to Cho and Kreps, QJE 1987
� Fix a equilibrium (DL*, DH*) and consider a deviation from this equilibrium to D’. If the deviation never benefits type x (even if it induces the most favorable beliefs by the capital market) but can benefit type y, then the deviation was played by type y
� The intuitive criterion:
� The intuitive criterion still does not determine the beliefs everywhere
=
=
wo
DD
DD
B p
p
/'
*by dominated is 1
*
*
γ
γ
Corporate Finance 17
The set of pooling equilibria under the intuitive criterion
D
L
DC
LV +=
2 H
DC
LV +=
2
2
HV =
2
LV =
V
D* D**
poolingequil.
VV ˆ= The intuitive criterion eliminates all pooling equil.
Corporate Finance 18
Conclusions
� The only equilibrium which survives the intuitive criterion is DL* = 0 and DH* = D*
� This equilibrium is the Pareto undominated separating equilibrium and it is called the “Riley outcome”
� Debt can be used as a signal of high cash flow
� The debt of type H:
� D*↑ when L↑, H-L↑, and C↓, D* is independent of γ!
( )C
LLHD
H
L
DC
L
2*
22
−=⇒=+
Leland and Pyle, JF 1977
“Informational asymmetries, financial
structure, and financial intermediation”
Corporate Finance 20
The model� The timing:
� Cash flow is x ~ [0,∞), with Ex = xT and Var(x)=σ2, where T ∈ {L,H}, xL < xH
� T is private info. to the firm
� The capital market believes that T = H with prob. γ
� The entrepreneur’s expected utility:
An entrepreneur sell a fraction 1-α of the firm to outside equityholders
Stage 0 Stage 2Stage 1
The capital market observes 1-α and updates the belief about the firm’s type
Cash flow is realized
( ) ( )VxWWVarb
EWWEU TTTT αα −+=×−= 1)(2
Corporate Finance 21
The full information case� The variance of WT:
� The entrepreneur’s expected utility:
� Under full info., V = xT:
⇒ α* = 0 ⇒ The entrepreneur will sell the entire firm
� Why? Because the entrepreneur is risk-averse and the capital market is risk-neutral
( ) ( )( )( )( ) 222
21
σαα
αα
=−=
−−+=
T
TT
xxE
EWVxEWVar
( ) ( ) 32144 344 21)(
22
21
xVarEW
TT
bVxWEU
T
σααα ×−−+=
( ) 22
2σα×−=
bxWEU TT
Corporate Finance 22
Asymmetric information
� αT* is the equity participation of type T
� B* is the prob. that the capital market assigns to the firm being of type T
� Perfect Bayesian Equilibrium (PBE), (αH*, αL*,B*):� αH* and αL* are optimal given B*
� B* is consistent with the Bayes rule
Corporate Finance 23
Separating equilibria: αH* ≠ αL*� The belief function:
� In a separating equil., αL* = 0 because type L cannot boost V by keeping equity
� αL* = 0 ⇒ B*(0) = 0 ⇒ V(0) = xL
=
=
=
wo
B L
H
/'
*0
*1
*
γαααα
Corporate Finance 24
Separating equilibrium� The IC of H:
� The IC of L:
� Indifference of type T between V and α:
( ){
0 when H of Payoff
firm theof fraction a keeps he when H of Payoff
22
21
=
≥−−+
αα
σααα LH xb
Vx4444 34444 21
{( )
4444 34444 21firm theof fraction a keeps he when L of Payoff
22
0 when L of Payoff
21
αα
σαααb
Vxx LL −−+≥
=
{( )
αασ
αα
σααα
αα
−×+
−−
=⇒−−+=
=
12121
22
V valuea inducesit if firm theof fraction a keeping when Payoff
22
0 ifPayoff
bxxV
bVxx TL
TL
4444 34444 21
Corporate Finance 25
The set of separating equilibria
α
αασ−
×+=12
22b
xV L
HxV =
LxV =
V
α* α**
sep. equil.
αασ
αα
−×+
−−
=121
22bxx
V HL
( )2
* 2σ
αb
xxZZZZ LH −≡++−=
Corporate Finance 26
The Riley outcome� Using L’s IC, the ownership share of type H is:
� α* increases with Z which � increases with the extent of asymmetric info., xH-xL� decreases with risk aversion, b� decreases with the variance of cash flows, σ2
� Using H’s IC:
� α** < 1 iff 2(xH–xL)< bσ2; otherwise, type H prefers every α > 0 over α = 0
provided that he convinces outsiders that the firm’s type is H
{( ) { ( )2*
21
2
H is typehis that believesmarket theandfirm theof fraction a keeps he when L of Payoff
22
0 when L of Payoff
++−=⇒−−+=−
=
ZZZb
xxx
b
xx
HLL
LH
σαα
ασααα4444 34444 21
{( ) ( )
2
H is typehis that believesmarket theandfirm theof fraction a keeps he when H of Payoff
22
0 when H of Payoff
2**
21
σασααα
αα
b
xxbxxx LH
HHL
−=⇒−−+=
=4444 34444 21