Corporate Finance:Agency models of capital structure
Yossi SpiegelRecanati School of Business
Jensen and Meckling, JFE 1976
The agency cost of outside equity
Corporate Finance 3
The investment model
The timing:
V’(I)>0>V”(I) and V(0) = 0
V’(0) = and V’() = 0 (interior sol’n)
The entrepreneur’s payoff: U(I) = V(I) + (R-I)
Period 1 Period 2
The firm is establishedby an entrepreneur whohas R dollars and needsto invest I
The investment yieldsV(I), R-I is consumed as perks
V(I)
I
Corporate Finance 4
Internal financing F.O.C for the entrepreneur’s problem:
.1*)('01')(' IVIVIU
V’(I)
I
1
I*
Corporate Finance 5
Internal financing The entrepreneur’s payoff at the
optimum:
But what if I* > R? In that case the entrepreneur must have external financing
RIIVIU ***)(
Corporate Finance 6
Debt financing The entrepreneur issues debt with face value D to raise
I*- R upfront The entrepreneur’s payoff:
U(I,D) is maximized at I* (the firm invests optimally) Debt is safe: ex post the firm has V(I*) > I* > I*-R
D* = I*-R The entrepreneur’s payoff:
Perks
anteex flowCash
postex Payoff
*),( IRIRDIVDIU
RIIV
IRIRDIVDIU
**)(
*****)*,(
Corporate Finance 7
Equity financing with commitment The entrepreneur issues equity with equity
participation 1- (the entrepreneur keeps ) and commits to invest I*
To raise I*-R:
The entrepreneur’s payoff:
.*)(
*1***)()1(*
IVRIRIIV
E
RIIV
IREIVIU
**)(
*****)*,(Perkspayoffpost Ex
Corporate Finance 8
Equity financing without commitment The entrepreneur issues equity with
equity participation 1-, but cannot commit to invest I*
After receiving E, but before choosing I, the entrepreneur’s payoff:
Perkspayoffpost Ex
),( IREIVIU
Corporate Finance 9
Equity financing without commitment F.O.C for the entrepreneur’s problem:
We get underinvestment (more outside equity) I* (more
underinvestment)
V’(I)
I
1
I** I*
V’(I)
.1*)*('01'),(' IVIVIU
Corporate Finance 10
The agency cost of outside equity The entrepreneur’s payoff:
The entrepreneur’s payoff with commitment:
By revealed preferences (and since I** < I*):
The entrepreneur bears the cost of underinvestment (outside investors break even)
RIIVRIIVIV
RIEIVIU
***)*(***)*(1*)*(
******)*,*(
RIIVIU **)()*,(
RIIVRIIV ***)*(**)(
Corporate Finance 11
The optimal choice of How does affect U(I**,)?
The entrepreneur will raise up to the point where
E**=0 at = 0 (since then I**=0) and at = 1 E** is inverse U-shaped
0**1*)*(')*,*(
)()(
/1
IIVIU
**
**1**E
IVRI
Corporate Finance 12
The entrepreneur’s budget constraint:
The relevant sol’n is with the maximal
The optimal choice of
**
E**
R
I**
Corporate Finance 13
The effort model
The timing:
V’(e)>0>V”(e) and V(0) = 0
V’(0) = and V’() = 0 (interior sol’n)
The entrepreneur’s payoff: U(e) = V(e) - e
Period 1 Period 2
The firm is establishedby an entrepreneur whohas R dollars and needsto invest I and exerteffort, e
The effort yieldsV(e), the cost of effortis e (no perks)
V(e)
e
Corporate Finance 14
Internal financing F.O.C for the entrepreneur’s problem:
1*)('01')(' eVeVeU
V’(e)
e
1
e*
Corporate Finance 15
Internal financing The entrepreneur’s payoff at the
optimum:
But what if I > R?
Debt financing: debt is safe so D*=I-R
***)( eeVeU
Corporate Finance 16
Equity financing without commitment The entrepreneur issues equity with
equity participation 1-
After receiving E, but before choosing e, the entrepreneur’s payoff is:
eeVeU ),(
Corporate Finance 17
F.O.C for the entrepreneur’s problem:
We get underinvestment
Equity financing without commitment
V’(e)
e
1
e** e*
V’(e)
1*)*('01'),(' eVeVeU
Corporate Finance 18
The agency cost of outside equity The entrepreneur’s payoff:
The entrepreneur’s payoff with commitment:
By revealed preferences (and since e** < e*):
The entrepreneur bears the cost of underinvestment (outside investors break even)
****)*,*( eeVeU
***)*(**)( eeVeeV
**)*,( eeVeU
Corporate Finance 19
The optimal choice of How does affect U(e**,)?
The entrepreneur will raise up to the point where
E**=0 at = 0 (since then e**=0) and at = 1 E** is inverse U-shaped
0**1*)*('*)*()*,*(
0
eeVeVeU
**
**1E
eVRI
Corporate Finance 20
The entrepreneur’s budget constraint:
The relevant sol’n is with the maximal
The optimal choice of
**
E**
R
I
Jensen and Meckling, JFE 1976
The asset substitution problem
Corporate Finance 22
A simple example Consider a box with two sealed envelopes: one with $100
and the with $0
You can either pick one envelop from the box or receive $70 for sure – what would you do?
Now suppose you owe someone $50 out of your gains
If you take $70, your payoff is $70 - $50 = $20
If you pick one of the sealed envelops then you either have $100 - $50 = $50, or you pick the empty envelop and your payoff is 0 because you cannot pay the $50 your expected payoff is ($0+$50)/2 = $25
The lottery is better even though its NPV is only $50
Corporate Finance 23
The model Two projects: Safe project with return Z Risky project with return X ~ [0, )
The firm has debt with face value D
The management is perfect agent for equityholders – the agency problem is between equityholders and debtholders
Corporate Finance 24
Payoffs under the two projects Equityholders’ payoff with the risky project:
Equityholders’ payoff with the safe project:
The firm surely chooses the risky project if Z ≤ D assume that Z > D. Hence
0,DZMaxYS
D
R XdFDXY )(
DZYS
Corporate Finance 25
Comparing the two projects The safe project is better iff:
Properties of XC(D):
D
CRS XdFDXDDXZYY )()(
D
C
DFDFXdFD
DX )()(11)(1)(
XXXdFX C ˆ)()0(0
Corporate Finance 26
Choice of projects with leverage XC(D) > for all D > 0
A leveraged firm prefers the risky project even when it is inefficient
safe project is efficient
X )(DX C
X
safe project is chosen when D>0
risky project is efficient
risky project is chosen when D>0
Myers, JFE 1977
The debt overhang problem
Corporate Finance 28
The model
The timing:
In period 1 it is common knowledge that X~[0, ) Absent debt, the firm invests iff X ≥ I. The value
of the firm:
Period 1 Period 2
The firm is establishedby an entrepreneur who issues debt with face value D
Period 1.5
I
XdFIXV )()0(
The entrepreneur learnsthe return, X, from a project that costs I anddecides whether to invest
X is realized anddebt is paid
Corporate Finance 29
Illustrating – All-equity firm
XX1
Xf (X)
If(x)
I
E(0)
Corporate Finance 30
Short-term debt (due at period 1.5) Suppose debt has to be paid before it is
time to invest
If X – I ≥ D, the firm will invest
If X – I < D, the firm will not invest and will go bankrupt. The debtholders will invest provided that X ≥ I
The firm invests iff X ≥ I investment is efficient
Corporate Finance 31
The value of the firm with short-term debt The value of debt:
The value of equity:
The total value of the firm:
ID
ID
I
XDdFXdFIXDB )()()(
ID
XdFIDXDE )()(
I
ID
ID
I
XdFIX
XdFIXXdFIXDV
)(
)()()(
Corporate Finance 32
Illustrating the debt overhang problem
XX1
Xf (X)
If(x)
D+I
B(D)
E(D)
(D+I)f(x)
I
Corporate Finance 33
Long-term debt Suppose debt has to be paid after it is
time to invest
If X – I ≥ D, the firm will invest
If X – I < D, the firm will not invest and will go bankrupt. The debtholders get a firm with no investment opportunities
Corporate Finance 34
The value of the firm with long-term debt The value of debt:
The value of equity:
The total value of the firm:
The value is lower than under short-term debt. This is the debt overhang problem
ID
XDdFDB )()(
ID
XdFIDXDE )()(
ID
XdFIXDV )()(
Corporate Finance 35
Illustrating the debt overhang problem
XX1
Xf (X)
If(x)
D+I
B(D)
E(D)
(D+I)f(x)
I
Loss
Berkovitch and Kim, JF 1990
Overinvestment and underinvestment
Corporate Finance 37
The model
The timing:
g’(I)>0>g”(I) and g(0) = 0
g’(0) = and g’() = 0 (interior sol’n)
z~[0, )
Period 0 Period 2
The firm is establishedby an entrepreneur who issues debt with face value D
The cash flow X isrealized and the firminvests I
Period 1
Investment yields zg(I), D is due
g(I)
I
Corporate Finance 38
Excess funds: X > I Cash flow at the end of period 2:
The firm is solvent iff
The expected value of equity:
IXIzg )(
)()( 1 Ig
XIDzzDIXIzg
1
)()()(1z
zdFIDXIzgDE
Corporate Finance 39
Illustrating – excess funds
z1
(X-I)f(z)
z1
E1(D)
(zg(I)+X-I)f(z)
Df(z)
Corporate Finance 40
Investment with excess funds F.O.C for investment:
Rewriting:
0)(1)()('
)(1)(')(
1
1
1
1
zFzzdFIg
zdFIzgIDE
z
z
1
1
|1
)(1
)(
1)('
1
zzzE
zF
zzdFIg
z
Corporate Finance 41
Deficit: X < I The firm covers the deficit by issuing
extra equity Cash flow at the end of period 2: zg(I) The firm is solvent iff
The expected value of equity:
)()( 2 Ig
DzzDIzg
2
deficit cover the tofunds Extra
2 )()()(z
XIzdFDIzgDE
Corporate Finance 42
Illustrating – deficit
z1
(X-I)f(z)
z2
E2(D)
zg(I)f(z)
Df(z)
Corporate Finance 43
Investment with deficit F.O.C for investment:
Rewriting:
01)()('
1)()(')(
2
2
2
z
z
zzdFIg
zdFIzgIDE
2
)(
1)('
z
zzdFIg
Corporate Finance 44
The value of the firm with excess funds
IXIgz
zdFIXIzg
zDdFzdFIXIzg
zdFIDXIzgDV
z
z
z
)(ˆ
)()(
)()()(
)()()(
0
Debt
0
Equity
1
1
1
1
Corporate Finance 45
The value of the firm with deficit
The value of the firm is exactly as in the case of excess funds
IXIgz
XIzdFIzg
zDdFzdFIzgXIzdFDIzgDVz
z
z
)(ˆ
)()(
)()()()()()(
0
Debt
0
Equity
2
2
2
2
Corporate Finance 46
Efficient investment F.O.C for investment:
Excess funds:
Deficit:
zIgIgz
IDV
ˆ1)('01)('ˆ)(
zzzdF
Ig
z
ˆ1
)(
1)('
2
zzzzE
zF
zzdFIg
z
ˆ1
|1
)(1
)(
1)('1
1
1
Corporate Finance 47
Comparison
Ig’(I)
z1
1|1
zzzE
2
)(
1
z
zzdF
I2 I1I*
Excess funds overinvestment Deficit underinvestment