Part 4
Chapter 11
Yulin
[email protected]
Department of Finance,
Xiamen University



Main line:
1 A partial-equilibrium one-period
model
2 A general intertemporal equilibrium
model of the asset market, includes
three models(model 1 is based on a
constant interest rate assumption,
model 2 is a no-riskless-asset case,
model 3 is the general model).
Ⅰ A partial-equilibrium oneperiod model

We follows the warrant pricing approach
used in Chapter 7, that is, investors
choose among three assets: the warrant,
the stock of the firm and a riskless asset
to form optimal portfolios which
maximize their expected utility.
Consider an economy made up of only one
firm with current value V  t , and there exists a
“representative man” acts so as to maximize
the expected utility of wealth at the end of a
period of length, that
is,
max Et U V (t   )  …… ①
V (t   )
 Define a random variable Z by Z 
V (t )
and assume its probability distribution P( Z , ) is
known at present, more importantly,



P ( Z ,  ) is
independent of the particular
structure of the firm, this is consistent
with the MM(Modigliani-Miller) theorem.
Define Fi V ,  as the current value of the
i th type of security issued by the firm.
The different types of securities are
distinguishable by their terminal value
Fi VZ ,0. For a debt issue(i=1),
F1 VZ ,0  min( B,VZ )
…②





Because each of the securities appears
separately in the market, so:
n
n
V   Fi V ,   and VZ   Fi VZ ,  
1
1
Define wi  Fi V ,  V , so we can rewrite
① as a maximization under constraint:
n

 n
Fi VZ , 0  




max  EtU V  wi
  1   wi  

w
Fi V ,  
1



 1

 …③
i



The corresponding first-order conditions
are:  F VZ , 0   n F VZ , 0   
Et  i
U  V  wi i
  
Fi V ,   
 Fi V , 
 1
This can be rewritten in terms of utilprob distributions Q as:


0
 Fj VZ , 0 
Fi VZ , 0 
dQ  
dQ  exp( )
0
Fi V , 
Fj V , 
…④
dQ 


U   ZV  dP  Z , 
Where
0 U   ZV  dP  Z ,  and exp  is a
new multiplier related to  .
dQ is independent of the functions Fi by
the assumption that the value of the
firm is independent of its capital
structure, so ④ is a set of integral
equations linear in the Fi , and we can
rewrite ④ as





Fi V ,   exp     Fi VZ ,0 dQ  Z , 
…*
Suppose the firm issues just one type of
security--equity, then
F1 V ,   V , and , F1 VZ ,0  VZ
0
Substituting in ④, we have

exp     ZdQ  Z , 
0

From ④, we can see that the expected
return on all securities in util-prob space
must be equated. If U was linear, then
dQ=dP and ④ would imply the result
for the risk-neutral case. Hence, the
util-prob distribution is the distribution
of returns adjusted for risk.
Some examples




Example 1:
Firm issues two types of securities, debt
and equity with current value F1 V ,  and
F2 V , respectively. From ② and ④, we
have ⑤:
B /V


F1 V ,   exp     ZVdQ  Z ,    BdQ  Z ,  
 0

B /V
 exp    B  exp    
B /V
0
 B  ZV  dQ  Z , 
Suppose 0  B V or dQ  Z ,   0 for 0  Z  B V
then F1 V ,   exp    B as B V  0 .
 In the limit, the debt becomes riskless,
so  will be replaced by r. Another
useful form of ⑤ is



F1 V ,   exp  r   ZVdQ  Z ,     ZV  B  dQ  Z ,  
 0

B /V

 V  exp  r  

ZV  B  dQ  Z , 

B /V

Since in equilibrium V  F1 V ,   F2 V , 




So, F2 V ,   exp  r  B /V  ZV  B  dQ  Z ,  . This is
identical to the warrant pricing equation
derived in Chapter 7.
This equation can also be derived directly
from the terminal value of equity
F2 VZ ,0  max  0,VZ  B  in the same way as
debt.


Example 2:
Firm’s capital structure made up from
three types of securities: debt, equity(N
shares outstanding with current price
per share of S, i.e. F2 V ,   NS ), warrants
(exercise price is S ). Assume there are
n warrants outstanding with current
market value per warrant of W,


i.e. F3 V ,   nW . Because the warrant is a
junior security to the debt, the current
value of the debt will be the same as in
the first example. The current value of
the equity will be
 /V
F2 V ,   exp  r  [   ZV  B  dQ  Z ,  
B /V
N

n N
  ZV  nS  B  dQ  Z , ]

 /V
…⑥
Where   NS  B.
 Rewrite ⑥ as

F2 V ,   exp  r  [   ZV  B  dQ  Z , 
B /V

…⑦


 n N 

nS  1 
  ZV  B   dQ  Z , ]



/
V
n N
N 



N exp  r   
n

 V  F1 V ,  
nS

ZV

B

 dQ  Z , 



/
V
n N
N


N




In equilibrium, F3 V ,   V  F1 V ,   F2 V ,  .
So from ⑦ we have

n
F3 V ,  
exp  r    ZV    dQ  Z , 
⑧
 /V
n N
If we define normalized price of the
firm as
y
V


V n  N 
 NS  B   n  N 



And define the normalized price of a
F3 V ,  
warrant as w 
, then ⑧ can be
n  n  N 
rewritten as

w  y,   exp  r    Zy  1 dQ  Z ,  
 /V
which is of the same form as equation
(7.24).


Example 3:
Firm’s capital structure contains two
securities:convertible bonds with a total
terminal claim on the firm of either B
dollars or alternatively the bonds can be
exchanged for a total of n shares of
equity; and N shares of equity with
current price per share of S dollars.
So, F VZ , 0   max /V0, min VZ  B, NVZ n  N  , and
F2 V ,   exp  r   VZ  B  dQ  Z , 
B /V

2
N

n N


/V
VZdQ  Z ,  
Where  is determined to be  n  N  B
B /V
F1 V ,   V  F2 V ,   exp  r  [  VZdQ  Z , 
0


n
  BdQ  Z ,   
B /V
n N

 VZ    dQ  Z , ]
/V
n
.

By inspection of this equation, we have
the well-known result that the value of
a convertible bond is equal to its value
as a straight bond plus a warrant with
exercise price S  B n .


Example 4:
A “dual” fund: it issues two types of
securities to finance its assets: namely,
capital shares(equity) which are entitled
to all the accumulated capital gains(in
excess of the fixed terminal payment);
and income-shares(a type of bond)
which are entitled to all the ordinary


income in addition to a fixed terminal
payment.
Let  be the instantaneous fixed
proportion of total asset value earned
as ordinary income, V be the current
asset value of the fund and Z the total
return on the fund.

Let F V ,  be the current value of the
income shares with terminal claim on
the fund of B dollars plus all interest
and dividends earned, F V ,  be the
current value of the capital shares.
So, from definition, we have
F2 VZ , 0   max 0, exp    VZ  B 
1
2


And F2 V ,   exp    r      /V VZ   dQ  Z , 
 Where   B exp(  ) .
 /V
F1 V ,   V  F2 V ,   exp  r { VZdQ  Z , 
0



 /V

BdQ  Z ,   

 /V
VZ 1  exp     dQ  Z , 
The current value of the capital shares
can be less than the current net asset
value of the capital shares, defined to

be V-B, because

F2 V ,   exp    r      VZ   dQ  Z , 
 /V
 exp    r     

 /V

VZdQ  Z ,   exp    V
If exp    V  V  B , that is, V  B
then, F V ,   V  B .
2
1  exp     
Ⅱ A general intertemporal
equilibrium model

Consider an economy with K consumers
–investors and n firms with current
value Vi , i  1, , n .Each consumer acts so
as to

max E0  U k C k  s  , s  ds  B k W k T k  , T k 
0


k

Define Ni t  Pi t   Vi t , where Ni t is the
number of shares and Pi  t  is the price
per share at time t.

Assume that expectations about the
dynamics of the prices per share in the
futures are the same for all investors
and can be described by the stochastic
differential equation:
dPi
  i dt   i dZ i , i  1,..., n
Pi


Further assume that one of the n assets
(the nth one) is an instantaneously
riskless asset with instantaneous
return r  t  :
dr  f  r, t  dt  g  r , t  dq
For simplicity, we assume that i and  i
are functions only of r  t  .

From Ni t  Pi t   Vi t , dVi  Ni dPi  dNi  Pi  dPi 
divide both side by Vi and substitute
for dPi Pi ,then
dVi
dNi
  i dt   i dZi 
1  i dt   i dZi 
Vi


Ni
The accumulation equation for the kth
investor can be written as
n
k
k
k dPi
dW   wi W
  y k  C k  dt
…⑨
1
Pi


k
w
Where y is his wage income and i
is
the fraction of his wealth invested in the
ith security.
So, his demand for the ith security d ik
can be written as
k
dik  wikW k  N ik Pi

k
N
Where i is the number of shares of the
ith security demanded by investor k.

Substituting for dPi Pi , we can rewrite ⑨
as n
n


dW k   wik i  r   r  W k dt   wikW k i dZ i   y k  C k  dt
n
1
 1

k
k


From the budget constraint, W   Ni Pi
1
and from ⑨, wen have
 y k  C k  dt   dNik  Pi  dPi 
1
i.e. the net value of shares purchased
must equal the value of savings from
wage income.

According to Chapter 4 and 5, the necessary
optimality conditions for an individual who
acts to maximize his expected utility are
k
k
k
k
k
0  max
(
U
C
,
t

J
W
,
r
,
t

J




3
2 f
k
k
(C ,w )
 m k
1 k 2
 k
k
k 
 J    wi  i  r   r  W   y  C    J 22 g

 1
 2
m
2
1 k m m k k
 J11  wi w j  ij W k  J12k   ir wikW k ) (10)
2
1
1
1
subject to J k W k , r , T k   B k W k , T k  .
k
1


From (10), we can derive m+1 first-order
conditions
k
k
k
k
 0  U1  C , t   J1 W , r , t 
(11)
m
k
k
k
k
k
0

J


r

J
w
W


J



1
i
11 
j
ij
12 ir , i  1,..., m (12)
1
 Equation (12) can be solved explicitly for
the demand functions for each risky
security as
m
m

dik  Ak  vij  j  r   H k  vij jr , i  1,..., m
1
1



where the vij are the elements of the
inverse of the instantaneous variancecovariance matrix of returns    ij  ,
Ak   J1k J11k and H k   J12k J11k .
Applying the Implicit Function Theorem
to (11), we have Ak  U U C k  0
1
C k
H 
r
k
11
W k
C k
 0
k
W

The aggregate demands Di are
Di   dik  A vij  j  r   H  vij jr , i  1,..., m
K
m
m
(13)
1
1
1
 If it is assumed that the asset market is
always
in equilibrium, then
n
n
  N i Pi   Di  M ,where M is the total
1
1
value of all assets. So, from ⑨
n
n
K
1
1
1
dM   N i dPi   dN i  Pi  dPi    dW k
dPi K k
  Di
   y  C k  dt
Pi
1
1
n

(14)

Let PM be the price per “share” of the
market portfolio and N be the number
of “shares”, i.e. M  NPM . Then,
dM  NdPM  dN  PM  dPM 

N and PM are defined by
n
NdPM   N i dPi
1
n
dN  PM  dPM    dN i ( Pi  dPi )
1
From  y k  C k  dt   dNik  Pi  dPi  and
1
K
k
dN i  1 dN i , then
K
dN  PM  dPM     y k  C k  dt
1
 And from (13), we get
n
dPi
NdPM   Di

(15)
Pi
1
 Define wi  Ni Pi M  Di M .Divide equation
(14) by M and substitute for dPi Pi
n


We canm rewrite (15) as
m
dPM


   w j  j  r   r  dt   w j j dZ j

PM
1
 1


And from
this we can determine
m
 M   w j  j  r   r
1
m
dPM dPi
 iM dt 
  w j ij dt , i  1,..., m
PM Pi
1
dPM dPM
2
 M dt 

PM PM
m
dPM
 Mr dt 
dr   w j jr dt
PM
1
(16)
From (13), we can solve for the yields
on individual risky assets in matrixvector form:
1
H
  r  D   r

(17)
A
A
 Since in equilibrium, Di  wi M , it can be
rewritten in scalar form as
M m
H
M
H
 i  r   w j ij   ir   iM   ir , i  1,..., m (18)

A
1
A
A
A



Multiplying both side by wi and summing
from 1 to m, we have
M 2 H
 M  r   M   Mr
(19)
A
A
Hence, from (18) and (19), if we know
the equilibrium prices, then the
equilibrium expected yields of the risky
assets and the market portfolio can be
determined. The equilibrium interest
rate can be determined from (11).
Model 1: A constant interest
rate assumption

With a constant interest rate, the ratios
of an investor’s demands for risk assets
are the same for all investors. Hence, the
“mutual-fund” or separation theorem
holds, and all optimal portfolios can be
represented as a linear combination of
any two distinct efficient portfolios(we
can choose them to be the market
portfolio and the riskless asset).




By combining (18) and (19) we have
 iM
 i  r  2  M  r  (20)
M
With a slightly different interpretation of
the variables, (20) is the equation for
the Security Market Line of CAPM.
If it is assumed that the wiare constant
over time, then from (16), PM is lognormally distributed.



We can integrate the stochastic process
for dPi to get conditional on Pi  t   Pi

1 2

Pi  t     Pi exp  i   i    i Zi  t ;  
2 

t  
where Zi  t;   t dZi is a normal variate
with zero mean and variance  .
Similarly, we can integrate (16) to get

1 2
PM  t     PM exp   M   M
2



    M X  t ;  


t 



where X t;   t 1 wj j dZi  Mis a normal
variate with zero mean and variance  .
Define the variables
 Pi  t     
1 2
pi  t     log 
    i   i     i Z i  t ; 
2 
 Pi  t   
 PM  t     
1 2 
pM  t     log 
    M   M     M X  t ; 
2

 PM  t   
m



Then consider the ordinary least-squares
i
p
t

r


p
t

r










regression i
i  M
 i t , if
Model 1 is the true specification, then the
following must hold:
 iM
1
i
2 12
2
2
 i  2 ;  i    i M   i ;  t  1  iM   iYi  t; 
M
2
Yi  t; is a normal variate with zero mean
and a covariance with the market return
of zero.


Reconsider the first example where firm’s
capital structure consists of two securities:
equity and debt, and it is assumed that
the firm is enjoined from the issue or
purchase of securities prior to the
redemption date of the debt namely, dNi  0
dV dP

  dt   dZ
So,
(21)
V
P



Let D  t;  be the current value of the
debt with  years until maturity and with
redemption value at that time of B.
Let F V ,  be the current value of equity
and the dynamics of the return on
equity can be written as
dF
(22)
  e dt   e dZ
F
 iM
 i  r  2  M  r 
M




(20)
Like every security in the economy, the
equity of the firm must satisfy (20) in
equilibrium, hence,
 e M
e  r 
 M  r 
(23)
2
M
1
2
By Ito’s lemma, dF  FV dV  F d  2 FVV  dV 
substitute dV from (21), we get:
1

dF    2V 2 FVV  VFV  F  dt   VFV dZ (24)
2


1 2 2

dF    V FVV  VFV  F  dt   VFV dZ
2

dF
  e dt   e dZ
F






Comparing (24) with (22), it must be that
1 2 2
 e F   V FVV  VFV  F (25)
2
 e F   VFV
(26)
And also, in equilibrium,
 M
 r 
 M  r 
2
(27)
M
Substituting for and e from (23) and
(27), we have the Fundamental Partial






Differential Equation of Security Pricing
1 2 2
0   V FVV  rVFV  F  rF
(28)
2
subject to the boundary condition
F V ,0  max  0,V  B 
The solution is

F V ,   exp  r   VZ  B d   Z ,  (29)
BV
Z is a log-normally distributed random
variable with mean exp  r and variance
of log  Z  2 , and d is the log-normal
density function.



(28) is the Fundamental Partial Differential
Equation of Asset Pricing because all the
securities in the firm’s capital structure
must satisfy it. And each securities are
distinguished by their terminal claims.
Equation (29) can be rewritten in general
form as

F V ,   exp  r   F VZ ,0d   Z ,  (30)
0

Although (30) is a kind of discounted
expected value formula, one should not
infer that the expected return on F is r.
From (23),(26) and (27), the expected
return on F can be written as
FV V
e  r 
  r 
F


(28) was derived by Black and Scholes (1973)
under the assumption of market equilibrium
when pricing option contracts, but it actually
holds without this assumption.
Consider a two-asset portfolio which contains
the firm as one security and any one of the
security in the firm’s capital structure as the
other.
Let P be the price per unit of this portfolio,
 the fraction of the total portfolio’s value
invested in the firm and 1  the fraction in
the particular security chosen from the
firm’s capital structure.
 So,

dP
dV
dF

 (1   )
     e    e  dt      e    e  dZ
P
V
F



Suppose  is chosen such that     e    e  0
Then the portfolio will be perfectly hedged
and the instantaneous return on the
portfolio will be certain(equal to r),that is,
e
   e   e  r . So  e  r    r  .

Then, as was done previously, we can
arrived at (28). Nowhere was the market
equilibrium assumption needed!
pi  t   r   i  pM  t   r    i   ti


Remarks:
Although the value of the firm follows a simple
dynamic process with constant parameters, the
individual component securities follow more
complex processes with changing expected
returns and variances. Thus, in empirical
examinations using a regression, if one were to
use equity instead of firm values, systematic
biases would be introduced.
Model 2: The “no riskless
asset” case

If there exists uncertain inflation and there
are no future markets in consumption
goods or other guaranteed “purchasing
power” securities available, there will be no
perfect hedge against future price changes,
i.e. no riskless asset exists.




Follow the same procedure as in section
11.4, we can derive analogous equilibrium
conditions, namely,
M
(31)
 i   iM  G, i  1,..., n
A
M 2
M   M  G
A
(32)
The nth security must satisfy (31) in
M
equilibrium  n   Mn  G
(33)
A
 iM
 i  r  2  M  r 
M

Solve M A and G in (32),(33) and
substitute them into (31), we have
 Mi   Mn
 M2   Mi
i  2
M  2
 n , i  1,..., m
 M   Mn
  M   Mn
(34)

In a similar fashion to the analysis in
Model 1, we get the Fundamental
Partial Differential Equation for Security
1 2 2
Pricing 0   V FVV  VFV  F   F ,where
2
    n   Mn M     Mn 
2
M
2
M
If security n is a zero-beta security, i.e.
  Mn  0,then    n ,and (34) can be
rewritten as

i  iM  1  i n , i  1,..., m


2




where i Mi M .
Model 3: The general model




In this model, the interest rate varies
stochastically over time.
In section 11.4, we have
M
H
 i  r   iM   ir , i  1,..., m (35)
A
A
M 2 H
 M  r   M   Mr
A
A
(36)


k  r 
M
H
So,  m  r   mM   mr
A
A
(37)
Solve (36) and (37) for M A and H A ,
and substitute them into (35), we have
 mr Mk   Mm kr


Q
 M  r  
 M2  kr   Mr Mk
Q
m  r 
(38)
2
Q


M  mr   Mr Mm and k  1,..., m  1
where


Theorem 11.1(Three “Fund” Theorem)
Given n assets satisfying the conditions of the
model in section 11.4, there exist three
portfolios (“mutual funds”) constructed from
these n assets such that all risk-averse
individuals, who behave to maximize their
expected utilities, will be indifferent from
these three funds.Further, a possible choice
for the three funds is the market portfolio,
the riskless asset, and a portfolio which is
(instantaneously)perfectly correlated with
changes in the interest rate.

Since in this model, the interest rate
varies stochastically, we can determine
the term structure from this model, and
nowhere in the model is it necessary to
introduce concepts such as liquidity,
transactions costs,time horizon or habit
to explain the existence of a term
structure.



Suppose there exists a security(mth
security) is perfectly correlated with
changes in the interest rate, and its
dynamics are described by
dPm
  m dt   m dq
(39)
Pm
and from dr  f  r, t  dt  g  r, t  dq, we have
 Mm   Mm M  m    Mr M  m
 m  Mr m  Mr m
   Mr M  r


r
r
g
 mr   m g

Let P  r , be the price of a discounted
loan which pays a dollar at time  in the
future when the current interest rate is r.
Then the dynamics of P can be written
dP
  dt    dq . And also we have
as

P
 M    M   M      Mr M      Mr M  r



 Mr 
r
 Mr 

g

r
 r   g
k  r 
 mr Mk   Mm kr
Q





 M  r  
 M2  kr   Mr Mk
Q
m  r 
(38)
Set k   in (38) and substitute
 Mm ,  mr ,  M ,   r , we have

  r 
 m  r 
(40)
m
By Ito’s lemma, and  must satisfy
1 2
0  g Prr  fPr  P   P
2
Pr g
 
P
(41)




So, given  , (41) can be solved to
determine P  r,  and hence the term
structure of interest rate.
Suppose the Expectations Hypothesis
holds, then   r for all  and the term
structure is completely determined by
1 2
0  g Prr  fPr  P  rP
(42)
2
subject to P  r,0  1.
k  r 
 mr Mk   Mm kr
Q

 M  r  
 M2  kr   Mr Mk
Q
m  r 
(38)
From (40), it must be that in
equilibrium  m  r ,and the equilibrium
condition (38) simplifies to
 r2 Mk   Mr kr
k  r 
 M  r 
2
2
2
 M  r   Mr
 r2  Mk M  k   Mr  kr M  k r2

 M  r 
2
2
2
2
2
 M  r   Mr M  r
 k   Mk   Mr  kr 

 M  r 
2
 M 1   Mr 


And further if we assume f and g are
constants, we have the explicit solution
for (42):
2

f 2 g 3
P  r ,   exp  r     
2
6 

that P   as    ,which
Note
all reasonable.
is not at

Just in the similar way as Model 1 and
Model 2,we can derive the Fundamental
Equation of Security Pricing as
1 2 2
1 2
0   V FVV  g Frr   g FrV  rVFV  fFr  F  rf
2
2

subject to an appropriate boundary
condition F V , r,0 .


Remark 1: the model does not allow for
nonhomogeneous expectations, non-serially
independent preferences, or transactions
costs.
Remark 2:the fundamental assumption in this
model is continuous-time assumption.If the
model were formulated in discrete time, then
the results derived in this chapter no longer
hold.
The end
Thanks!