Literature Part2
報告者：陳政岳
1
Literature
• Black and Cox (1976)
• Leland (1994)
• Leland and Toft (1996)
2
Black and Cox Approach
• Black and Cox simplicity assume that it is a consol
bond, which pays continuously coupon rate c.
Assume that r > 0 and the payout rate  is equal to
zero in dV
t
  r    dt   V dZ t
*
Vt
v ：a critical level of the value of the firm, below
which no more equity can be sold.
v ：fixed bankruptcy level.
 *：the optimal default time,
 *  inf t  0：Vt  v*  inf t  0：Vt  v*.
3
Black and Cox Approach (cont.)


u

u
• By ODE for the pricing function
V 
of a consol bond：
1 2 2 


V  uVV  rVuV  c  ru  0
• If
vc
2
r




v 
v 
c   1 c    c

u Vt     v  v  Vt  1      v  
r 
r 
r   Vt  
 Vt 


2r
 2

4
Black and Cox Approach (cont.)
• It is the interest of the stockholders to select
the bankruptcy level in such a way that the
value of the debt, D Vt   u Vt 
• D Vt  is minimized.
• the value of the firm’s equity is maximized.

 c   v  
v  
E Vt   Vt  D Vt   Vt   1      v   
Vt  
 r   Vt  




v 
c
 Vt  1  qt   v  qt , qt   
r
 Vt 
5
Black and Cox Approach (cont.)
• Solving
dD V 
dv
0
V v
c 
c
 v 

r  1 r   2
*

2
v 
c  v  
c 1
Dc  t   1      v     
r   Vt  
r Vt
 Vt 



c 1 c
c
  

r Vt  r r   2
2


c

 r  2
2

c
 

v

v
r



 c 1

r Vt


c

 r  2
2

 1




6
Leland’s Approach
• 基本假設如 Black and Cox Approach.
• 增加了Bankruptcy costs 和 Tax benefits.
7
Leland’s Approach
• VB:the level of asset value which bankruptcy is
declared.
• VB:bankruptcy cost, 0   . 1
• By solving ODE, we get 2r
C 
C  V  
D V    1   VB    (7)

2
r

V 
BC V   VB  
 VB 
r   VB 
 2r
2
(9)
8
Leland’s Approach
•
•
•
C
CV 
TB V         
r
r  VB 
 2r
2
(11)
v V   V  TB V   BC V 
2r 2
  2 r  2 
V  
C
V
  VB  
 V   1  
r   VB 

 VB 


(12)
E V   v V   D V 
C 
C
 V 
 V  1     1     VB   
r 
r
  VB 
(13)
2 r
2
9
Leland’s Approach (cont.)
• Unprotected debt:
Assumed that the bankruptcy occurs, the value
of the equity may hit zero.
Equity can choose any barrier.
• Protected debt:
Assumed that the bankruptcy occurs, the value
of the VB is VB  D0
• Protected versus Unprotected Debt: Potential
Agency Problems
10
Unprotected Debt
• The bankruptcy occurs whenever the value of
the equity hits barrier, no matter what is the
chosen level of the default triggering barrier.
• It is the interest of the stockholders to select the
bankruptcy level in such a way that the value of
the debt is minimized, and thus the value of the
firm’s equity is maximized.
dE
 0.
• Assumed V  VB is such that dV
V V
C
Solving v*  1     2 (14)
B
r
2
11
Unprotected Debt (cont.)
1   C


1   C

X

1 X r   2
2
Observe that the asset value VB
r
• a) is proportional to the coupon C;
• b) is independent of the current asset value, V;
• c) decreases as the corporate tax rate,  ,
increases;
• d) is independent of bankruptcy costs, ;
• e) decreases as the risk-free interest rate, r, rises;
• f) decreases with increases in the riskiness of
the firm,  2 .
12
Unprotected Debt (cont.)
• The maximal value of
the firm’s equality is
X

C  C 
*
E Vt   V  1    1    m 
(17)
r   V 

• The minimal value of the firm’s debt equals
X

C
C 
*
(15).
D Vt   1    k 
r   V  
X
m  1    X / r 1  X  
1  X 
h  1  X   1    X /   m
k  1  X  1   1    X  m
2r
X 2

13
Unprotected Debt (cont.)
• A. The Comparative Statics of Debt Value
(D(V))
• B. Yield Spreads: The risk Structure of
Interest Rates
• C. The Comparative Statics of Firm Value
(v(V)) and Equity Value (E(V))
14
A. The Comparative Statics of Debt
Value (D(V)):
• D(V) is eventually decreasing as the coupon
rises, implies that debt value reaches a
maximum, Dmax V  , for a finite coupon, Cmax V  .

C  C
• Differentiating equation(15) D V   r 1   V  k 


with respect to C, setting the resulting
equation equal to zero and solving for C:
1/ X
Cmax V   V 1  X  k  (19).
• Substituting (19) into equation (15),
X
*
t
V
Dmax V  
r
 1
  X1
1  
 Xk 1  X   X  


15
A. The Comparative Statics of Debt
Value (D(V)): (cont.)
• The debt capacity of a firm is proportional
to asset value, V, and falls with increases
2
in firm risk, , and bankruptcy costs,  .
Debt capacity rises with increase in the
corporate tax rate,  , and the risk-free rate,
r.

X

2

V 2r
Dmax V    2 k 2 r
r 

 2r 
1  2 
  
2

1 
 2r 




m  1    X / r 1  X  
1  X 
k  1  X  1   1    X  m
16
A. The Comparative Statics of Debt
Value (D(V)): (cont.)
17
B. Yield Spreads: The risk Structure
of Interest Rates
Junk bond yields spreads may actually decline
when firm riskiness increases.
Investment-grade debt yield spreads increase when18
firm risk risk rise.
C. The Comparative Statics of Firm
Value (v(V)) and Equity Value (E(V))
19
Optimal Leverage with Unprotected
Debt
• Consider the coupon rate, C, which
maximizes the total value, v, of the firm.
• Differentiation equation (16) with respect to
C, setting the derivative equal to zero and 1

*
solving for the optimal couponC V   V 1  X  h X .

1

1
V
X




D V  
1

X
h
1

k
1

X
h







r 
1


 
 X 
*
X

v V   V 1    
1

X
h




1  X  
r





*
R* V  
r
1  X  h 

1  X  h  k
m
V V   V 

 h 
*
B
1
X

m  1    X / r 1  X  
X
1  X 
h  1  X   1    X /   m
k  1  X  1   1    X  m
X
2r
2
20
Optimal Leverage with Unprotected
Debt (cont.)
• The optimal leverage rations of riskier
firms will always be less than those of less
risky firms, as can be seen by observing
the maximal firm values in Figure7.
• The optimal coupon is a U-shaped
function of firm riskiness, as illustrated in
Figure8.
21
Optimal Leverage with Unprotected
Debt (cont.)
22
Protected Debt
• P : outstanding debt when bankruptcy is
triggered by the firm’s assets falling
beneath the principal value of debt
• D0 : the principal coincides with the market
value of the debt
2r
C 
C  V  
• From equation (7) D V   r  1   VB  r   VB 
with VB  D0 , we can write the value of
protected debt as a function of the value of
assets , V0 : D V   C  1    D V   C   V 
2
2 r
2
0
0
0
r

0
0
r   D V0  
23
Protected Debt (cont.)
• Figures 10 and 11 illustrate the behavior of
protected debt value as the coupon and
leverage change.
• Increased firm risk or a higher risk-free
interest rate always lowers debt value.
24
Protected Debt (cont.)
25
Protected Debt (cont.)
When there are no bankruptcy costs(   0 )
• a) Protected debt is riskless and pays the
risk-free rate r.
• b) For any C, the value of the tax shield
with protected debt is less than the tax
shield with unprotected debt.
• c) For any C, the bankruptcy-triggering
value of assets, VB , for protected debt
excess the VB for unprotected debt.
C
VB  P  D0 V0   ( protected )
r
VB 
1    C
r
1    C (unprotected )
X

26
1 X r   2
2
Protected Debt (cont.)
• When bankruptcy costs are zero. Asset
value falling to the principal value, then
bankruptcy is declared and, debtholders
receive their full principal value.
C
VB  P  D0 V0  
r
• When bankruptcy costs exist. For a given
coupon, protected debt have a lesser
value than unprotected debt.
27
Optimal Leverage with Protected
Debt (cont.)
28
Optimal Leverage with Protected
Debt
• Using a simple procedure to find the
*
C
coupon, , that maximizes the total value,
v, of the firm with protected debt. Figure
13, compared with Figure 7 illustrate that
maximal firm value occurs at lower
leverage.
29
Optimal Leverage with Protected
Debt (cont.)
30
Optimal Leverage with Protected
Debt (cont.)
• a) Optimal leverage for protected debt is
substantially less than for unprotected debt.
• b) The interest rate paid at the optimum
leverage is less for protected debt, even
when bankruptcy costs are positive.
• c) The maximum value of the firm is less when
protected debt is used.
• d) The maximal benefits of unprotected over
protected debt increase as:
-Corporate taxes increase
-Interest rates are higher
-Bankruptcy costs are lower
31
Protected versus Unprotected Debt:
Potential Agency Problems
32
Leland and Toft Approach
• As in Merton (1974), Black and Cox (1976),
and Brennan and Schwartz (1978), the
firm asset value V follow a continuous
diffusion process dV    V , t     dt   dz
V
where  V , t  is the total expected rate of
return on asset value V;  is the constant
fraction of value paid out to security
holders; and dz is the increment of a
standard Brownian motion.
34
Leland and Toft Approach
• Endogenous bankruptcy with a stationary
Debt Structure
• Applications
35
Endogenous bankruptcy with a
stationary Debt Structure
• A. A Stationary Debt Structure
• B. Bankruptcy: determining the Bankruptcy
-Triggering Asset Level VB
36
A. A Stationary Debt Structure
• Consider firm continuously sell a constant
amount of new debt with maturity of T years
and it will redeem at par upon maturity.
• The firm remains solvent, at any time s the
total outstanding debt principal have a
uniform distribution in the interval (s,s+T)
37
A. A Stationary Debt Structure
(cont.)
P
• p  : new bond principal issued rate, and
T
P is the total principal value of all
outstanding bonds.
C
• c  : coupon rate, and C is total coupon
T paid.
• Total debt payments are time-independent
P
and equal to C  .
T
38
A. A Stationary Debt Structure
(cont.)
•
D V ;VB , T  :
the total value of debt, when debt
of maturity T is issued.

•   t   ,   1   : bond holders receive the
T
remaining
value.
T
D V ;VB , t    d V ;VB , t dt
t 0
 
C 
C   1  e  rT
C
   P  
 I T     1   VB   J T 
r 
r   rT
r
 
(7)
T
1
where I T    e  rt F  t dt
T 0
1 T
39
J (T )   G  t dt
T 0
A. A Stationary Debt Structure
(cont.)
• v: the total market value of the firm, and
following Leland (1994)
x
x


V 
V 
C
1      VB   (8)
v V ;VB   V 
r   VB  
VB 



E V ;VB , T   v V ;VB   D V ;VB , T  (9)
其中x = a + z,

 r 
a
  2 
 
 2 

2
,z

 a

2

2
12
 2r 

2
2
40
B. Bankruptcy: determining the
Bankruptcy-Triggering Asset Level
VB
• Solving the equation E V ;VB , T 
,
V
we can solve equation (10) for
Cx
 C  A
 AP

B



 

r  rT
r

 rT
VB 
1   x  1    B
0
V VB
(11)


 
2
2e

n  z T  
n  a T    z  a 
 T
 T
A  2ae  rT N a T  2 zN z T
 rT
2

B    2z  2
z T





2
1

N
z

T

n
z

T

z

a




z 2T
 T

41
B. Bankruptcy: determining the
Bankruptcy-Triggering Asset Level
VB (cont.)
1   Cx
• When T  , it can be shown that VB 
1 x r
as Lealnd(1994)
• VB depend on the maturity of debt.
• In contrast, the models of the VB with flowbased bankruptcy (Kim, Ramaswamy, and
Sundaresan (1993) and Ross(1994)) or with
a positive net worth covenant (Longstaff and
Schwartz (1995)) is independent of debt
maturity.
42
,
B. Bankruptcy: determining the
Bankruptcy-Triggering Asset Level
VB (cont.)
dE V V
B


P 1   VB
  1    C  
  VB  dt
T
T


  1    C  p  dt   d VB ;VB , T    VB  dt (12)
• The l.h.s. of equation (12) is the change in
the equity value at V  VB .
• The r.h.s. is the additional cash flow
required from equity holders for current
debt service.
• Tax deductibility is lost whenever V falls
below some value VT , where VT  VB
43
B. Bankruptcy: determining the
Bankruptcy-Triggering Asset Level
VB (cont.)
 C  A
 AP
 B 
 
r rT
 rT
VB   
(13)
 C

1 x 
    1    B
 rVT





2
2e

n  z T  
n  a T    z  a 
 T
 T
A  2ae  rT N a T  2 zN z T
 rT
2

B    2z  2
z T





2
1

n z T   z  a   2
 N z T 
z T
 T

44
B. Bankruptcy: determining the
Bankruptcy-Triggering Asset Level
VB (cont.)
• When VT  VB ,VB in equation (13) will exceed
VB in equation (11).
45
Applications
• Considered the loss of tax deductibility
and cash flow (coupon).
• When V  VT , VT  C.
• When VB  V  VT ,equity holders willingly
contribute to the firm to avoid default, and
a dilution of their holders resulting from
stock issuance.
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Ⅲ. Applications (cont.)
A. Optimal Leverage
B. Debt Value and Debt Capacity
C. The Term Structure of Credit Spreads
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A. Optimal Leverage
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A. Optimal Leverage (cont.)
• Barclay and Smith (1995) find a positive
correlation between leverage and debt
maturity.
• For any maturity, the optimal leverage ratio
falls, when firm risk and bankruptcy costs
increase.
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A. Optimal Leverage (cont.)
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B. Debt Value and Debt Capacity
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B. Debt Value and Debt Capacity
(cont.)
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B. Debt Value and Debt Capacity
(cont.)
• Debt capacity falls, as debt maturity less,
volatility or bankruptcy cost rise.
• Low and intermediate leverages show
“humped” market value: newly-issued
bonds and about-to-be-redeemed bonds
sell at par; bonds with remaining maturity
leverage between 0 and T sell above par.
• The hump is more pronounced for greater
leverage levels and longer maturities.
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C. The Term Structure of Credit
Spreads
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C. The Term Structure of Credit
Spreads (cont.)
• Credit spreads : high leverage levels,
spreads are high, but decrease as
issuance maturity T increases beyond 1
year.
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